:: JORDAN1A semantic presentation

3 = (2 * 1) + 1
;

then Lemma1: 3 div 2 = 1
by NAT_1:def 1;

1 = (2 * 0) + 1
;

then Lemma2: 1 div 2 = 0
by NAT_1:def 1;

theorem Th1: :: JORDAN1A:1
canceled;

theorem Th2: :: JORDAN1A:2
canceled;

theorem Th3: :: JORDAN1A:3
canceled;

theorem Th4: :: JORDAN1A:4
for b1, b2 being natural number holds
( b2 > 0 & b1 mod b2 = 0 implies b1 div b2 = b1 / b2 ) by PEPIN:68;

theorem Th5: :: JORDAN1A:5
for b1, b2 being natural number holds
( b2 > 0 implies (b1 |^ b2) div b1 = (b1 |^ b2) / b1 ) by PEPIN:69;

theorem Th6: :: JORDAN1A:6
for b1 being real number
for b2 being natural number holds
not ( 0 < b2 & 1 < b1 & not 1 < b1 |^ b2 ) by PEPIN:70;

theorem Th7: :: JORDAN1A:7
for b1 being real number
for b2, b3 being natural number holds
not ( b1 > 1 & b2 > b3 & not b1 |^ b2 > b1 |^ b3 ) by PEPIN:71;

theorem Th8: :: JORDAN1A:8
for b1 being non empty TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2 is connected & b4 is_a_component_of b3 & b2 meets b4 & b2 c= b3 implies b2 c= b4 ) by GOBOARD9:6;

definition
let c1 be FinSequence;
func Center c1 -> Nat equals :: JORDAN1A:def 1
((len a1) div 2) + 1;
coherence
((len c1) div 2) + 1 is Nat
;
end;

:: deftheorem Def1 defines Center JORDAN1A:def 1 :
for b1 being FinSequence holds Center b1 = ((len b1) div 2) + 1;

theorem Th9: :: JORDAN1A:9
for b1 being FinSequence holds
( not len b1 is even implies len b1 = (2 * (Center b1)) - 1 )
proof end;

theorem Th10: :: JORDAN1A:10
for b1 being FinSequence holds
( len b1 is even implies len b1 = (2 * (Center b1)) - 2 )
proof end;

registration
cluster non empty being_simple_closed_curve compact non horizontal non vertical Element of K18(the carrier of (TOP-REAL 2));
existence
ex b1 being Subset of (TOP-REAL 2) st
( b1 is compact & not b1 is vertical & not b1 is horizontal & b1 is being_simple_closed_curve & not b1 is empty )
proof end;
cluster non empty compact horizontal Element of K18(the carrier of (TOP-REAL 2));
existence
ex b1 being Subset of (TOP-REAL 2) st
( b1 is compact & not b1 is empty & b1 is horizontal )
proof end;
cluster non empty compact vertical Element of K18(the carrier of (TOP-REAL 2));
existence
ex b1 being Subset of (TOP-REAL 2) st
( b1 is compact & not b1 is empty & b1 is vertical )
proof end;
end;

theorem Th11: :: JORDAN1A:11
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in N-most b1 implies b2 `2 = N-bound b1 )
proof end;

theorem Th12: :: JORDAN1A:12
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in E-most b1 implies b2 `1 = E-bound b1 )
proof end;

theorem Th13: :: JORDAN1A:13
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in S-most b1 implies b2 `2 = S-bound b1 )
proof end;

theorem Th14: :: JORDAN1A:14
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in W-most b1 implies b2 `1 = W-bound b1 )
proof end;

theorem Th15: :: JORDAN1A:15
for b1 being Subset of (TOP-REAL 2) holds BDD b1 misses b1
proof end;

theorem Th16: :: JORDAN1A:16
canceled;

theorem Th17: :: JORDAN1A:17
for b1 being Point of (TOP-REAL 2) holds b1 in Vertical_Line (b1 `1 )
proof end;

theorem Th18: :: JORDAN1A:18
for b1, b2 being real number holds |[b1,b2]| in Vertical_Line b1
proof end;

theorem Th19: :: JORDAN1A:19
for b1 being real number
for b2 being Subset of (TOP-REAL 2) holds
( b2 c= Vertical_Line b1 implies b2 is vertical )
proof end;

theorem Th20: :: JORDAN1A:20
canceled;

theorem Th21: :: JORDAN1A:21
for b1, b2 being Point of (TOP-REAL 2)
for b3 being real number holds
( b1 `1 = b2 `1 & b3 in [.(proj2 . b1),(proj2 . b2).] implies |[(b1 `1 ),b3]| in LSeg b1,b2 )
proof end;

theorem Th22: :: JORDAN1A:22
for b1, b2 being Point of (TOP-REAL 2)
for b3 being real number holds
( b1 `2 = b2 `2 & b3 in [.(proj1 . b1),(proj1 . b2).] implies |[b3,(b1 `2 )]| in LSeg b1,b2 )
proof end;

theorem Th23: :: JORDAN1A:23
for b1, b2 being Point of (TOP-REAL 2)
for b3 being real number holds
( b1 in Vertical_Line b3 & b2 in Vertical_Line b3 implies LSeg b1,b2 c= Vertical_Line b3 )
proof end;

theorem Th24: :: JORDAN1A:24
for b1, b2 being Subset of (TOP-REAL 2) holds
not ( b1 meets b2 & not proj2 .: b1 meets proj2 .: b2 )
proof end;

theorem Th25: :: JORDAN1A:25
for b1 being real number
for b2, b3 being Subset of (TOP-REAL 2) holds
( b2 misses b3 & b2 c= Vertical_Line b1 & b3 c= Vertical_Line b1 implies proj2 .: b2 misses proj2 .: b3 )
proof end;

theorem Th26: :: JORDAN1A:26
for b1 being closed Subset of (TOP-REAL 2) holds
( b1 is Bounded implies proj2 .: b1 is closed ) by JCT_MISC:22;

theorem Th27: :: JORDAN1A:27
for b1 being Subset of (TOP-REAL 2) holds
( b1 is Bounded implies proj2 .: b1 is bounded ) by JCT_MISC:23;

theorem Th28: :: JORDAN1A:28
for b1 being compact Subset of (TOP-REAL 2) holds proj2 .: b1 is compact by JCT_MISC:24;

theorem Th29: :: JORDAN1A:29
for b1 being Point of (TOP-REAL 2) holds north_halfline b1 = { b2 where B is Point of (TOP-REAL 2) : ( b2 `1 = b1 `1 & b2 `2 >= b1 `2 ) } by TOPREAL1:37;

theorem Th30: :: JORDAN1A:30
for b1 being Point of (TOP-REAL 2) holds north_halfline b1 = { |[(b1 `1 ),b2]| where B is Element of REAL : b2 >= b1 `2 } by TOPREAL1:38;

theorem Th31: :: JORDAN1A:31
for b1 being Point of (TOP-REAL 2) holds east_halfline b1 = { b2 where B is Point of (TOP-REAL 2) : ( b2 `1 >= b1 `1 & b2 `2 = b1 `2 ) } by TOPREAL1:39;

theorem Th32: :: JORDAN1A:32
for b1 being Point of (TOP-REAL 2) holds east_halfline b1 = { |[b2,(b1 `2 )]| where B is Element of REAL : b2 >= b1 `1 } by TOPREAL1:40;

theorem Th33: :: JORDAN1A:33
for b1 being Point of (TOP-REAL 2) holds south_halfline b1 = { b2 where B is Point of (TOP-REAL 2) : ( b2 `1 = b1 `1 & b2 `2 <= b1 `2 ) } by TOPREAL1:41;

theorem Th34: :: JORDAN1A:34
for b1 being Point of (TOP-REAL 2) holds south_halfline b1 = { |[(b1 `1 ),b2]| where B is Element of REAL : b2 <= b1 `2 } by TOPREAL1:42;

theorem Th35: :: JORDAN1A:35
for b1 being Point of (TOP-REAL 2) holds west_halfline b1 = { b2 where B is Point of (TOP-REAL 2) : ( b2 `1 <= b1 `1 & b2 `2 = b1 `2 ) } by TOPREAL1:43;

theorem Th36: :: JORDAN1A:36
for b1 being Point of (TOP-REAL 2) holds west_halfline b1 = { |[b2,(b1 `2 )]| where B is Element of REAL : b2 <= b1 `1 } by TOPREAL1:44;

theorem Th37: :: JORDAN1A:37
for b1, b2 being Nat
for b3 being Go-board holds
( 1 <= b1 & b1 <= len b3 & 1 <= b2 & b2 <= width b3 implies b3 * b1,b2 in LSeg (b3 * b1,1),(b3 * b1,(width b3)) )
proof end;

theorem Th38: :: JORDAN1A:38
for b1, b2 being Nat
for b3 being Go-board holds
( 1 <= b1 & b1 <= len b3 & 1 <= b2 & b2 <= width b3 implies b3 * b1,b2 in LSeg (b3 * 1,b2),(b3 * (len b3),b2) )
proof end;

theorem Th39: :: JORDAN1A:39
for b1, b2, b3, b4 being Nat
for b5 being Go-board holds
( 1 <= b1 & b1 <= width b5 & 1 <= b2 & b2 <= width b5 & 1 <= b3 & b3 <= b4 & b4 <= len b5 implies (b5 * b3,b1) `1 <= (b5 * b4,b2) `1 )
proof end;

theorem Th40: :: JORDAN1A:40
for b1, b2, b3, b4 being Nat
for b5 being Go-board holds
( 1 <= b1 & b1 <= len b5 & 1 <= b2 & b2 <= len b5 & 1 <= b3 & b3 <= b4 & b4 <= width b5 implies (b5 * b1,b3) `2 <= (b5 * b2,b4) `2 )
proof end;

theorem Th41: :: JORDAN1A:41
for b1 being Nat
for b2 being Go-board
for b3 being non constant standard special_circular_sequence holds
( b3 is_sequence_on b2 & 1 <= b1 & b1 <= len b2 implies (b2 * b1,(width b2)) `2 >= N-bound (L~ b3) )
proof end;

theorem Th42: :: JORDAN1A:42
for b1 being Nat
for b2 being Go-board
for b3 being non constant standard special_circular_sequence holds
( b3 is_sequence_on b2 & 1 <= b1 & b1 <= width b2 implies (b2 * 1,b1) `1 <= W-bound (L~ b3) )
proof end;

theorem Th43: :: JORDAN1A:43
for b1 being Nat
for b2 being Go-board
for b3 being non constant standard special_circular_sequence holds
( b3 is_sequence_on b2 & 1 <= b1 & b1 <= len b2 implies (b2 * b1,1) `2 <= S-bound (L~ b3) )
proof end;

theorem Th44: :: JORDAN1A:44
for b1 being Nat
for b2 being Go-board
for b3 being non constant standard special_circular_sequence holds
( b3 is_sequence_on b2 & 1 <= b1 & b1 <= width b2 implies (b2 * (len b2),b1) `1 >= E-bound (L~ b3) )
proof end;

theorem Th45: :: JORDAN1A:45
for b1, b2 being Nat
for b3 being Go-board holds
not ( b1 <= len b3 & b2 <= width b3 & cell b3,b1,b2 is empty )
proof end;

theorem Th46: :: JORDAN1A:46
for b1, b2 being Nat
for b3 being Go-board holds
( b1 <= len b3 & b2 <= width b3 implies cell b3,b1,b2 is connected )
proof end;

theorem Th47: :: JORDAN1A:47
for b1 being Nat
for b2 being Go-board holds
not ( b1 <= len b2 & cell b2,b1,0 is Bounded )
proof end;

theorem Th48: :: JORDAN1A:48
for b1 being Nat
for b2 being Go-board holds
not ( b1 <= len b2 & cell b2,b1,(width b2) is Bounded )
proof end;

theorem Th49: :: JORDAN1A:49
for b1 being Nat
for b2 being non empty Subset of (TOP-REAL 2) holds width (Gauge b2,b1) = (2 |^ b1) + 3
proof end;

theorem Th50: :: JORDAN1A:50
for b1, b2 being Nat
for b3 being non empty Subset of (TOP-REAL 2) holds
not ( b1 < b2 & not len (Gauge b3,b1) < len (Gauge b3,b2) )
proof end;

theorem Th51: :: JORDAN1A:51
for b1, b2 being Nat
for b3 being non empty Subset of (TOP-REAL 2) holds
( b1 <= b2 implies len (Gauge b3,b1) <= len (Gauge b3,b2) )
proof end;

theorem Th52: :: JORDAN1A:52
for b1, b2, b3 being Nat
for b4 being non empty Subset of (TOP-REAL 2) holds
( b1 <= b2 & 1 < b3 & b3 < len (Gauge b4,b1) implies ( 1 < ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 & ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 < len (Gauge b4,b2) ) )
proof end;

theorem Th53: :: JORDAN1A:53
for b1, b2, b3 being Nat
for b4 being non empty Subset of (TOP-REAL 2) holds
( b1 <= b2 & 1 < b3 & b3 < width (Gauge b4,b1) implies ( 1 < ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 & ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 < width (Gauge b4,b2) ) )
proof end;

theorem Th54: :: JORDAN1A:54
for b1, b2, b3, b4 being Nat
for b5 being non empty Subset of (TOP-REAL 2) holds
( b1 <= b2 & 1 < b3 & b3 < len (Gauge b5,b1) & 1 < b4 & b4 < width (Gauge b5,b1) implies for b6, b7 being Nat holds
( b6 = ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 & b7 = ((2 |^ (b2 -' b1)) * (b4 - 2)) + 2 implies (Gauge b5,b1) * b3,b4 = (Gauge b5,b2) * b6,b7 ) )
proof end;

theorem Th55: :: JORDAN1A:55
for b1, b2, b3 being Nat
for b4 being non empty Subset of (TOP-REAL 2) holds
( b1 <= b2 & 1 < b3 & b3 + 1 < len (Gauge b4,b1) implies ( 1 < ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 & ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 <= len (Gauge b4,b2) ) )
proof end;

theorem Th56: :: JORDAN1A:56
for b1, b2, b3 being Nat
for b4 being non empty Subset of (TOP-REAL 2) holds
( b1 <= b2 & 1 < b3 & b3 + 1 < width (Gauge b4,b1) implies ( 1 < ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 & ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 <= width (Gauge b4,b2) ) )
proof end;

E30: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2 be Nat;
set c3 = Gauge c1,c2;
0 + 1 <= ((len (Gauge c1,c2)) div 2) + 1 by XREAL_1:8;
hence 1 <= Center (Gauge c1,c2) ;
4 <= len (Gauge c1,c2) by JORDAN8:13;
then 0 < len (Gauge c1,c2) ;
then (len (Gauge c1,c2)) div 2 < len (Gauge c1,c2) by INT_1:83;
then (len (Gauge c1,c2)) div 2 < len (Gauge c1,c2) by NEWTON:101;
then ((len (Gauge c1,c2)) div 2) + 1 <= len (Gauge c1,c2) by NAT_1:38;
hence Center (Gauge c1,c2) <= len (Gauge c1,c2) ;
end;

E31: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2, c3 be Nat;
set c4 = Gauge c1,c2;
assume E32: ( 1 <= c3 & c3 <= len (Gauge c1,c2) ) ;
E33: len (Gauge c1,c2) = width (Gauge c1,c2) by JORDAN8:def 1;
0 + 1 <= ((len (Gauge c1,c2)) div 2) + 1 by XREAL_1:8;
then E34: 0 + 1 <= Center (Gauge c1,c2) ;
Center (Gauge c1,c2) <= len (Gauge c1,c2) by Lemma30;
hence [(Center (Gauge c1,c2)),c3] in Indices (Gauge c1,c2) by E32, E33, E34, GOBOARD7:10;
end;

E32: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2, c3 be Nat;
set c4 = Gauge c1,c2;
assume E33: ( 1 <= c3 & c3 <= len (Gauge c1,c2) ) ;
E34: len (Gauge c1,c2) = width (Gauge c1,c2) by JORDAN8:def 1;
0 + 1 <= ((len (Gauge c1,c2)) div 2) + 1 by XREAL_1:8;
then E35: 0 + 1 <= Center (Gauge c1,c2) ;
Center (Gauge c1,c2) <= len (Gauge c1,c2) by Lemma30;
hence [c3,(Center (Gauge c1,c2))] in Indices (Gauge c1,c2) by E33, E34, E35, GOBOARD7:10;
end;

Lemma33: for b1 being Nat
for b2 being non empty Subset of (TOP-REAL 2)
for b3 being real number holds
( b1 > 0 implies (b3 / (2 |^ b1)) * ((Center (Gauge b2,b1)) - 2) = b3 / 2 )
proof end;

E34: now
let c1, c2 be Nat;
let c3, c4 be real number ;
assume E35: 0 <= c3 ;
E36: 0 < 2 |^ c1 by NEWTON:102;
assume c1 <= c2 ;
then 2 |^ c1 <= 2 |^ c2 by PREPOWER:107;
hence c3 / (2 |^ c2) <= c3 / (2 |^ c1) by E35, E36, XREAL_1:120;
end;

E35: now
let c1, c2 be Nat;
let c3, c4 be real number ;
assume ( 0 <= c3 & c1 <= c2 ) ;
then c3 / (2 |^ c2) <= c3 / (2 |^ c1) by Lemma34;
hence c4 + (c3 / (2 |^ c2)) <= c4 + (c3 / (2 |^ c1)) by XREAL_1:8;
end;

E36: now
let c1, c2 be Nat;
let c3, c4 be real number ;
assume ( 0 <= c3 & c1 <= c2 ) ;
then c3 / (2 |^ c2) <= c3 / (2 |^ c1) by Lemma34;
hence c4 - (c3 / (2 |^ c1)) <= c4 - (c3 / (2 |^ c2)) by REAL_1:92;
end;

theorem Th57: :: JORDAN1A:57
for b1, b2, b3, b4 being Nat
for b5 being non empty Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b5,b2) & 1 <= b3 & b3 <= len (Gauge b5,b4) & ( ( b2 > 0 & b4 > 0 ) or ( b2 = 0 & b4 = 0 ) ) implies ((Gauge b5,b2) * (Center (Gauge b5,b2)),b1) `1 = ((Gauge b5,b4) * (Center (Gauge b5,b4)),b3) `1 )
proof end;

theorem Th58: :: JORDAN1A:58
for b1, b2, b3, b4 being Nat
for b5 being non empty Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b5,b2) & 1 <= b3 & b3 <= len (Gauge b5,b4) & ( ( b2 > 0 & b4 > 0 ) or ( b2 = 0 & b4 = 0 ) ) implies ((Gauge b5,b2) * b1,(Center (Gauge b5,b2))) `2 = ((Gauge b5,b4) * b3,(Center (Gauge b5,b4))) `2 )
proof end;

E38: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2 be Nat;
let c3, c4 be real number ;
E39: 2 |^ c2 <> 0 by NEWTON:102;
thus c4 + (((c3 - c4) / (2 |^ c2)) * ((len (Gauge c1,c2)) - 2)) = c4 + (((c3 - c4) / (2 |^ c2)) * (((2 |^ c2) + 3) - 2)) by JORDAN8:def 1
.= (c4 + (((c3 - c4) / (2 |^ c2)) * (2 |^ c2))) + ((c3 - c4) / (2 |^ c2))
.= (c4 + (c3 - c4)) + ((c3 - c4) / (2 |^ c2)) by E39, XCMPLX_1:88
.= c3 + ((c3 - c4) / (2 |^ c2)) ;
end;

E39: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2, c3 be Nat;
set c4 = N-bound c1;
set c5 = S-bound c1;
set c6 = W-bound c1;
set c7 = E-bound c1;
set c8 = Gauge c1,c2;
assume [c3,1] in Indices (Gauge c1,c2) ;
hence ((Gauge c1,c2) * c3,1) `2 = |[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * (c3 - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * (1 - 2)))]| `2 by JORDAN8:def 1
.= (S-bound c1) - (((N-bound c1) - (S-bound c1)) / (2 |^ c2)) by EUCLID:56 ;

end;

E40: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2, c3 be Nat;
set c4 = N-bound c1;
set c5 = S-bound c1;
set c6 = W-bound c1;
set c7 = E-bound c1;
set c8 = Gauge c1,c2;
assume [1,c3] in Indices (Gauge c1,c2) ;
hence ((Gauge c1,c2) * 1,c3) `1 = |[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * (1 - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * (c3 - 2)))]| `1 by JORDAN8:def 1
.= (W-bound c1) - (((E-bound c1) - (W-bound c1)) / (2 |^ c2)) by EUCLID:56 ;

end;

E41: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2, c3 be Nat;
set c4 = N-bound c1;
set c5 = S-bound c1;
set c6 = W-bound c1;
set c7 = E-bound c1;
set c8 = Gauge c1,c2;
assume [c3,(len (Gauge c1,c2))] in Indices (Gauge c1,c2) ;
hence ((Gauge c1,c2) * c3,(len (Gauge c1,c2))) `2 = |[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * (c3 - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * ((len (Gauge c1,c2)) - 2)))]| `2 by JORDAN8:def 1
.= (S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * ((len (Gauge c1,c2)) - 2)) by EUCLID:56
.= (N-bound c1) + (((N-bound c1) - (S-bound c1)) / (2 |^ c2)) by Lemma38 ;

end;

E42: now
let c1 be non empty Subset of (TOP-REAL 2);
let c2, c3 be Nat;
set c4 = N-bound c1;
set c5 = S-bound c1;
set c6 = W-bound c1;
set c7 = E-bound c1;
set c8 = Gauge c1,c2;
assume [(len (Gauge c1,c2)),c3] in Indices (Gauge c1,c2) ;
hence ((Gauge c1,c2) * (len (Gauge c1,c2)),c3) `1 = |[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * ((len (Gauge c1,c2)) - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * (c3 - 2)))]| `1 by JORDAN8:def 1
.= (W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * ((len (Gauge c1,c2)) - 2)) by EUCLID:56
.= (E-bound c1) + (((E-bound c1) - (W-bound c1)) / (2 |^ c2)) by Lemma38 ;

end;

theorem Th59: :: JORDAN1A:59
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b2,1) implies ((Gauge b2,1) * (Center (Gauge b2,1)),b1) `1 = ((W-bound b2) + (E-bound b2)) / 2 )
proof end;

theorem Th60: :: JORDAN1A:60
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b2,1) implies ((Gauge b2,1) * b1,(Center (Gauge b2,1))) `2 = ((S-bound b2) + (N-bound b2)) / 2 )
proof end;

theorem Th61: :: JORDAN1A:61
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b5,b2) & 1 <= b3 & b3 <= len (Gauge b5,b4) & b4 <= b2 implies ((Gauge b5,b2) * b1,(len (Gauge b5,b2))) `2 <= ((Gauge b5,b4) * b3,(len (Gauge b5,b4))) `2 )
proof end;

theorem Th62: :: JORDAN1A:62
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b5,b2) & 1 <= b3 & b3 <= len (Gauge b5,b4) & b4 <= b2 implies ((Gauge b5,b2) * (len (Gauge b5,b2)),b1) `1 <= ((Gauge b5,b4) * (len (Gauge b5,b4)),b3) `1 )
proof end;

theorem Th63: :: JORDAN1A:63
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b5,b2) & 1 <= b3 & b3 <= len (Gauge b5,b4) & b4 <= b2 implies ((Gauge b5,b4) * 1,b3) `1 <= ((Gauge b5,b2) * 1,b1) `1 )
proof end;

theorem Th64: :: JORDAN1A:64
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b5,b2) & 1 <= b3 & b3 <= len (Gauge b5,b4) & b4 <= b2 implies ((Gauge b5,b4) * b3,1) `2 <= ((Gauge b5,b2) * b1,1) `2 )
proof end;

theorem Th65: :: JORDAN1A:65
for b1, b2 being Nat
for b3 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= b2 implies LSeg ((Gauge b3,b2) * (Center (Gauge b3,b2)),1),((Gauge b3,b2) * (Center (Gauge b3,b2)),(len (Gauge b3,b2))) c= LSeg ((Gauge b3,b1) * (Center (Gauge b3,b1)),1),((Gauge b3,b1) * (Center (Gauge b3,b1)),(len (Gauge b3,b1))) )
proof end;

theorem Th66: :: JORDAN1A:66
for b1, b2, b3 being Nat
for b4 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= b2 & 1 <= b3 & b3 <= width (Gauge b4,b2) implies LSeg ((Gauge b4,b2) * (Center (Gauge b4,b2)),1),((Gauge b4,b2) * (Center (Gauge b4,b2)),b3) c= LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),1),((Gauge b4,b2) * (Center (Gauge b4,b2)),b3) )
proof end;

theorem Th67: :: JORDAN1A:67
for b1, b2, b3 being Nat
for b4 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= b2 & 1 <= b3 & b3 <= width (Gauge b4,b2) implies LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),1),((Gauge b4,b2) * (Center (Gauge b4,b2)),b3) c= LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),1),((Gauge b4,b1) * (Center (Gauge b4,b1)),(len (Gauge b4,b1))) )
proof end;

theorem Th68: :: JORDAN1A:68
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( b1 <= b2 & 1 < b3 & b3 + 1 < len (Gauge b5,b1) & 1 < b4 & b4 + 1 < width (Gauge b5,b1) implies for b6, b7 being Nat holds
( ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 <= b6 & b6 < ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 & ((2 |^ (b2 -' b1)) * (b4 - 2)) + 2 <= b7 & b7 < ((2 |^ (b2 -' b1)) * (b4 - 1)) + 2 implies cell (Gauge b5,b2),b6,b7 c= cell (Gauge b5,b1),b3,b4 ) )
proof end;

theorem Th69: :: JORDAN1A:69
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( b1 <= b2 & 3 <= b3 & b3 < len (Gauge b5,b1) & 1 < b4 & b4 + 1 < width (Gauge b5,b1) implies for b6, b7 being Nat holds
( b6 = ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 & b7 = ((2 |^ (b2 -' b1)) * (b4 - 2)) + 2 implies cell (Gauge b5,b2),(b6 -' 1),b7 c= cell (Gauge b5,b1),(b3 -' 1),b4 ) )
proof end;

theorem Th70: :: JORDAN1A:70
for b1, b2 being Nat
for b3 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( b1 <= len (Gauge b3,b2) implies cell (Gauge b3,b2),b1,0 c= UBD b3 )
proof end;

theorem Th71: :: JORDAN1A:71
for b1, b2 being Nat
for b3, b4 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( b1 <= len (Gauge b3,b2) implies cell (Gauge b3,b2),b1,(width (Gauge b3,b2)) c= UBD b3 )
proof end;

theorem Th72: :: JORDAN1A:72
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & not north_halfline b3 meets L~ (Cage b2,b1) )
proof end;

theorem Th73: :: JORDAN1A:73
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & not east_halfline b3 meets L~ (Cage b2,b1) )
proof end;

theorem Th74: :: JORDAN1A:74
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & not south_halfline b3 meets L~ (Cage b2,b1) )
proof end;

theorem Th75: :: JORDAN1A:75
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & not west_halfline b3 meets L~ (Cage b2,b1) )
proof end;

Lemma49: for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
ex b3, b4 being Nat st
( 1 <= b3 & b3 <= len (Cage b2,b1) & 1 <= b4 & b4 <= width (Gauge b2,b1) & (Cage b2,b1) /. b3 = (Gauge b2,b1) * 1,b4 )
proof end;

Lemma50: for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
ex b3, b4 being Nat st
( 1 <= b3 & b3 <= len (Cage b2,b1) & 1 <= b4 & b4 <= len (Gauge b2,b1) & (Cage b2,b1) /. b3 = (Gauge b2,b1) * b4,1 )
proof end;

Lemma51: for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
ex b3, b4 being Nat st
( 1 <= b3 & b3 <= len (Cage b2,b1) & 1 <= b4 & b4 <= width (Gauge b2,b1) & (Cage b2,b1) /. b3 = (Gauge b2,b1) * (len (Gauge b2,b1)),b4 )
proof end;

theorem Th76: :: JORDAN1A:76
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
ex b3, b4 being Nat st
( 1 <= b3 & b3 < len (Cage b2,b1) & 1 <= b4 & b4 <= width (Gauge b2,b1) & (Cage b2,b1) /. b3 = (Gauge b2,b1) * 1,b4 )
proof end;

theorem Th77: :: JORDAN1A:77
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
ex b3, b4 being Nat st
( 1 <= b3 & b3 < len (Cage b2,b1) & 1 <= b4 & b4 <= len (Gauge b2,b1) & (Cage b2,b1) /. b3 = (Gauge b2,b1) * b4,1 )
proof end;

theorem Th78: :: JORDAN1A:78
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
ex b3, b4 being Nat st
( 1 <= b3 & b3 < len (Cage b2,b1) & 1 <= b4 & b4 <= width (Gauge b2,b1) & (Cage b2,b1) /. b3 = (Gauge b2,b1) * (len (Gauge b2,b1)),b4 )
proof end;

theorem Th79: :: JORDAN1A:79
for b1, b2, b3 being Nat
for b4 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Cage b4,b2) & 1 <= b3 & b3 <= len (Gauge b4,b2) & (Cage b4,b2) /. b1 = (Gauge b4,b2) * b3,(width (Gauge b4,b2)) implies (Cage b4,b2) /. b1 in N-most (L~ (Cage b4,b2)) )
proof end;

theorem Th80: :: JORDAN1A:80
for b1, b2, b3 being Nat
for b4 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Cage b4,b2) & 1 <= b3 & b3 <= width (Gauge b4,b2) & (Cage b4,b2) /. b1 = (Gauge b4,b2) * 1,b3 implies (Cage b4,b2) /. b1 in W-most (L~ (Cage b4,b2)) )
proof end;

theorem Th81: :: JORDAN1A:81
for b1, b2, b3 being Nat
for b4 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Cage b4,b2) & 1 <= b3 & b3 <= len (Gauge b4,b2) & (Cage b4,b2) /. b1 = (Gauge b4,b2) * b3,1 implies (Cage b4,b2) /. b1 in S-most (L~ (Cage b4,b2)) )
proof end;

theorem Th82: :: JORDAN1A:82
for b1, b2, b3 being Nat
for b4 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Cage b4,b2) & 1 <= b3 & b3 <= width (Gauge b4,b2) & (Cage b4,b2) /. b1 = (Gauge b4,b2) * (len (Gauge b4,b2)),b3 implies (Cage b4,b2) /. b1 in E-most (L~ (Cage b4,b2)) )
proof end;

theorem Th83: :: JORDAN1A:83
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds W-bound (L~ (Cage b2,b1)) = (W-bound b2) - (((E-bound b2) - (W-bound b2)) / (2 |^ b1))
proof end;

theorem Th84: :: JORDAN1A:84
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds S-bound (L~ (Cage b2,b1)) = (S-bound b2) - (((N-bound b2) - (S-bound b2)) / (2 |^ b1))
proof end;

theorem Th85: :: JORDAN1A:85
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds E-bound (L~ (Cage b2,b1)) = (E-bound b2) + (((E-bound b2) - (W-bound b2)) / (2 |^ b1))
proof end;

theorem Th86: :: JORDAN1A:86
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-bound (L~ (Cage b3,b1))) + (S-bound (L~ (Cage b3,b1))) = (N-bound (L~ (Cage b3,b2))) + (S-bound (L~ (Cage b3,b2)))
proof end;

theorem Th87: :: JORDAN1A:87
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-bound (L~ (Cage b3,b1))) + (W-bound (L~ (Cage b3,b1))) = (E-bound (L~ (Cage b3,b2))) + (W-bound (L~ (Cage b3,b2)))
proof end;

theorem Th88: :: JORDAN1A:88
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
not ( b1 < b2 & not E-bound (L~ (Cage b3,b2)) < E-bound (L~ (Cage b3,b1)) )
proof end;

theorem Th89: :: JORDAN1A:89
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
not ( b1 < b2 & not W-bound (L~ (Cage b3,b1)) < W-bound (L~ (Cage b3,b2)) )
proof end;

theorem Th90: :: JORDAN1A:90
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
not ( b1 < b2 & not S-bound (L~ (Cage b3,b1)) < S-bound (L~ (Cage b3,b2)) )
proof end;

theorem Th91: :: JORDAN1A:91
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b3,b2) implies N-bound (L~ (Cage b3,b2)) = ((Gauge b3,b2) * b1,(len (Gauge b3,b2))) `2 )
proof end;

theorem Th92: :: JORDAN1A:92
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b3,b2) implies E-bound (L~ (Cage b3,b2)) = ((Gauge b3,b2) * (len (Gauge b3,b2)),b1) `1 )
proof end;

theorem Th93: :: JORDAN1A:93
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b3,b2) implies S-bound (L~ (Cage b3,b2)) = ((Gauge b3,b2) * b1,1) `2 )
proof end;

theorem Th94: :: JORDAN1A:94
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( 1 <= b1 & b1 <= len (Gauge b3,b2) implies W-bound (L~ (Cage b3,b2)) = ((Gauge b3,b2) * 1,b1) `1 )
proof end;

Lemma62: for b1 being Point of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2) holds
( b1 in N-most b2 implies b1 in b2 )
by XBOOLE_0:def 3;

Lemma63: for b1 being Point of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2) holds
( b1 in E-most b2 implies b1 in b2 )
by XBOOLE_0:def 3;

Lemma64: for b1 being Point of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2) holds
( b1 in S-most b2 implies b1 in b2 )
by XBOOLE_0:def 3;

Lemma65: for b1 being Point of (TOP-REAL 2)
for b2 being Subset of (TOP-REAL 2) holds
( b1 in W-most b2 implies b1 in b2 )
by XBOOLE_0:def 3;

theorem Th95: :: JORDAN1A:95
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & b4 in (north_halfline b3) /\ (L~ (Cage b2,b1)) & not b4 `2 > b3 `2 )
proof end;

theorem Th96: :: JORDAN1A:96
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & b4 in (east_halfline b3) /\ (L~ (Cage b2,b1)) & not b4 `1 > b3 `1 )
proof end;

theorem Th97: :: JORDAN1A:97
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & b4 in (south_halfline b3) /\ (L~ (Cage b2,b1)) & not b4 `2 < b3 `2 )
proof end;

theorem Th98: :: JORDAN1A:98
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & b4 in (west_halfline b3) /\ (L~ (Cage b2,b1)) & not b4 `1 < b3 `1 )
proof end;

theorem Th99: :: JORDAN1A:99
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
( b4 in N-most b3 & b5 in north_halfline b4 & 1 <= b1 & b1 < len (Cage b3,b2) & b5 in LSeg (Cage b3,b2),b1 implies LSeg (Cage b3,b2),b1 is horizontal )
proof end;

theorem Th100: :: JORDAN1A:100
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
( b4 in E-most b3 & b5 in east_halfline b4 & 1 <= b1 & b1 < len (Cage b3,b2) & b5 in LSeg (Cage b3,b2),b1 implies LSeg (Cage b3,b2),b1 is vertical )
proof end;

theorem Th101: :: JORDAN1A:101
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
( b4 in S-most b3 & b5 in south_halfline b4 & 1 <= b1 & b1 < len (Cage b3,b2) & b5 in LSeg (Cage b3,b2),b1 implies LSeg (Cage b3,b2),b1 is horizontal )
proof end;

theorem Th102: :: JORDAN1A:102
for b1, b2 being Nat
for b3 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
( b4 in W-most b3 & b5 in west_halfline b4 & 1 <= b1 & b1 < len (Cage b3,b2) & b5 in LSeg (Cage b3,b2),b1 implies LSeg (Cage b3,b2),b1 is vertical )
proof end;

theorem Th103: :: JORDAN1A:103
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
( b3 in N-most b2 & b4 in (north_halfline b3) /\ (L~ (Cage b2,b1)) implies b4 `2 = N-bound (L~ (Cage b2,b1)) )
proof end;

theorem Th104: :: JORDAN1A:104
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
( b3 in E-most b2 & b4 in (east_halfline b3) /\ (L~ (Cage b2,b1)) implies b4 `1 = E-bound (L~ (Cage b2,b1)) )
proof end;

theorem Th105: :: JORDAN1A:105
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
( b3 in S-most b2 & b4 in (south_halfline b3) /\ (L~ (Cage b2,b1)) implies b4 `2 = S-bound (L~ (Cage b2,b1)) )
proof end;

theorem Th106: :: JORDAN1A:106
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3, b4 being Point of (TOP-REAL 2) holds
( b3 in W-most b2 & b4 in (west_halfline b3) /\ (L~ (Cage b2,b1)) implies b4 `1 = W-bound (L~ (Cage b2,b1)) )
proof end;

theorem Th107: :: JORDAN1A:107
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in N-most b2 & ( for b4 being Point of (TOP-REAL 2) holds
not (north_halfline b3) /\ (L~ (Cage b2,b1)) = {b4} ) )
proof end;

theorem Th108: :: JORDAN1A:108
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in E-most b2 & ( for b4 being Point of (TOP-REAL 2) holds
not (east_halfline b3) /\ (L~ (Cage b2,b1)) = {b4} ) )
proof end;

theorem Th109: :: JORDAN1A:109
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in S-most b2 & ( for b4 being Point of (TOP-REAL 2) holds
not (south_halfline b3) /\ (L~ (Cage b2,b1)) = {b4} ) )
proof end;

theorem Th110: :: JORDAN1A:110
for b1 being Nat
for b2 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 in W-most b2 & ( for b4 being Point of (TOP-REAL 2) holds
not (west_halfline b3) /\ (L~ (Cage b2,b1)) = {b4} ) )
proof end;