:: JORDAN22 semantic presentation

Lemma1: TOP-REAL 2 = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;

Lemma2: for b1 being real number
for b2 being Subset of (TOP-REAL 2) holds
not ( b1 in proj2 .: b2 & ( for b3 being Point of (TOP-REAL 2) holds
not ( b3 in b2 & proj2 . b3 = b1 ) ) )
proof end;

theorem Th1: :: JORDAN22:1
for b1 being Simple_closed_curve
for b2 being Nat holds (Upper_Appr b1) . b2 c= Cl (RightComp (Cage b1,0))
proof end;

theorem Th2: :: JORDAN22:2
for b1 being Simple_closed_curve
for b2 being Nat holds (Lower_Appr b1) . b2 c= Cl (RightComp (Cage b1,0))
proof end;

registration
let c1 be Simple_closed_curve;
cluster Upper_Arc a1 -> connected ;
coherence
Upper_Arc c1 is connected
proof end;
cluster Lower_Arc a1 -> connected ;
coherence
Lower_Arc c1 is connected
proof end;
end;

theorem Th3: :: JORDAN22:3
for b1 being Simple_closed_curve
for b2 being Nat holds
( (Upper_Appr b1) . b2 is compact & (Upper_Appr b1) . b2 is connected )
proof end;

theorem Th4: :: JORDAN22:4
for b1 being Simple_closed_curve
for b2 being Nat holds
( (Lower_Appr b1) . b2 is compact & (Lower_Appr b1) . b2 is connected )
proof end;

registration
let c1 be Simple_closed_curve;
cluster North_Arc a1 -> compact ;
coherence
North_Arc c1 is compact
proof end;
cluster South_Arc a1 -> compact ;
coherence
South_Arc c1 is compact
proof end;
end;

Lemma5: dom proj2 = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;

Lemma6: for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Nat holds 1 <= len (Gauge b1,b2)
proof end;

theorem Th5: :: JORDAN22:5
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Nat holds [1,1] in Indices (Gauge b1,b2)
proof end;

theorem Th6: :: JORDAN22:6
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Nat holds [1,2] in Indices (Gauge b1,b2)
proof end;

theorem Th7: :: JORDAN22:7
for b1 being non empty Subset of (TOP-REAL 2)
for b2 being Nat holds [2,1] in Indices (Gauge b1,b2)
proof end;

theorem Th8: :: JORDAN22:8
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
not ( b2 > b3 & [b1,b4] in Indices (Gauge b5,b3) & [b1,(b4 + 1)] in Indices (Gauge b5,b3) & not dist ((Gauge b5,b2) * b1,b4),((Gauge b5,b2) * b1,(b4 + 1)) < dist ((Gauge b5,b3) * b1,b4),((Gauge b5,b3) * b1,(b4 + 1)) )
proof end;

theorem Th9: :: JORDAN22:9
for b1, b2 being Nat
for b3 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
not ( b1 > b2 & not dist ((Gauge b3,b1) * 1,1),((Gauge b3,b1) * 1,2) < dist ((Gauge b3,b2) * 1,1),((Gauge b3,b2) * 1,2) )
proof end;

theorem Th10: :: JORDAN22:10
for b1, b2, b3, b4 being Nat
for b5 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
not ( b2 > b3 & [b1,b4] in Indices (Gauge b5,b3) & [(b1 + 1),b4] in Indices (Gauge b5,b3) & not dist ((Gauge b5,b2) * b1,b4),((Gauge b5,b2) * (b1 + 1),b4) < dist ((Gauge b5,b3) * b1,b4),((Gauge b5,b3) * (b1 + 1),b4) )
proof end;

theorem Th11: :: JORDAN22:11
for b1, b2 being Nat
for b3 being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
not ( b1 > b2 & not dist ((Gauge b3,b1) * 1,1),((Gauge b3,b1) * 2,1) < dist ((Gauge b3,b2) * 1,1),((Gauge b3,b2) * 2,1) )
proof end;

theorem Th12: :: JORDAN22:12
for b1 being Simple_closed_curve
for b2 being Nat
for b3, b4 being real number holds
not ( b3 > 0 & b4 > 0 & ( for b5 being Nat holds
not ( b2 < b5 & dist ((Gauge b1,b5) * 1,1),((Gauge b1,b5) * 1,2) < b3 & dist ((Gauge b1,b5) * 1,1),((Gauge b1,b5) * 2,1) < b4 ) ) )
proof end;

theorem Th13: :: JORDAN22:13
for b1 being Simple_closed_curve
for b2 being Nat holds
( 0 < b2 implies sup (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))) = sup (proj2 .: ((L~ (Cage b1,b2)) /\ (Vertical_Line (((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2)))) )
proof end;

theorem Th14: :: JORDAN22:14
for b1 being Simple_closed_curve
for b2 being Nat holds
( 0 < b2 implies inf (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))) = inf (proj2 .: ((L~ (Cage b1,b2)) /\ (Vertical_Line (((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2)))) )
proof end;

theorem Th15: :: JORDAN22:15
for b1 being Simple_closed_curve
for b2 being Nat holds
( 0 < b2 implies UMP (L~ (Cage b1,b2)) = |[(((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2),(sup (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))))]| ) by Th13;

theorem Th16: :: JORDAN22:16
for b1 being Simple_closed_curve
for b2 being Nat holds
( 0 < b2 implies LMP (L~ (Cage b1,b2)) = |[(((E-bound (L~ (Cage b1,b2))) + (W-bound (L~ (Cage b1,b2)))) / 2),(inf (proj2 .: ((L~ (Cage b1,b2)) /\ (LSeg ((Gauge b1,b2) * (Center (Gauge b1,b2)),1),((Gauge b1,b2) * (Center (Gauge b1,b2)),(len (Gauge b1,b2)))))))]| ) by Th14;

theorem Th17: :: JORDAN22:17
for b1 being Simple_closed_curve
for b2 being Nat holds
(UMP b1) `2 < (UMP (L~ (Cage b1,b2))) `2
proof end;

theorem Th18: :: JORDAN22:18
for b1 being Simple_closed_curve
for b2 being Nat holds
(LMP b1) `2 > (LMP (L~ (Cage b1,b2))) `2
proof end;

theorem Th19: :: JORDAN22:19
for b1 being Simple_closed_curve
for b2 being Nat holds UMP (Upper_Arc (L~ (Cage b1,b2))) in Upper_Arc (L~ (Cage b1,b2)) by JORDAN21:43;

theorem Th20: :: JORDAN22:20
for b1 being Simple_closed_curve
for b2 being Nat holds LMP (Lower_Arc (L~ (Cage b1,b2))) in Lower_Arc (L~ (Cage b1,b2)) by JORDAN21:44;

theorem Th21: :: JORDAN22:21
for b1 being Simple_closed_curve
for b2 being Nat holds
not ( 0 < b2 & ( for b3 being Nat holds
not ( 1 <= b3 & b3 <= len (Gauge b1,b2) & UMP (L~ (Cage b1,b2)) = (Gauge b1,b2) * (Center (Gauge b1,b2)),b3 ) ) )
proof end;

theorem Th22: :: JORDAN22:22
for b1 being Simple_closed_curve
for b2 being Nat holds
not ( 0 < b2 & ( for b3 being Nat holds
not ( 1 <= b3 & b3 <= len (Gauge b1,b2) & LMP (L~ (Cage b1,b2)) = (Gauge b1,b2) * (Center (Gauge b1,b2)),b3 ) ) )
proof end;

theorem Th23: :: JORDAN22:23
for b1 being Simple_closed_curve
for b2 being Nat holds
( 0 < b2 implies UMP (L~ (Cage b1,b2)) = UMP (Upper_Arc (L~ (Cage b1,b2))) )
proof end;

theorem Th24: :: JORDAN22:24
for b1 being Simple_closed_curve
for b2 being Nat holds
( 0 < b2 implies LMP (L~ (Cage b1,b2)) = LMP (Lower_Arc (L~ (Cage b1,b2))) )
proof end;

theorem Th25: :: JORDAN22:25
for b1 being Simple_closed_curve
for b2 being Nat holds
not ( 0 < b2 & not (UMP b1) `2 < (UMP (Upper_Arc (L~ (Cage b1,b2)))) `2 )
proof end;

theorem Th26: :: JORDAN22:26
for b1 being Simple_closed_curve
for b2 being Nat holds
not ( 0 < b2 & not (LMP (Lower_Arc (L~ (Cage b1,b2)))) `2 < (LMP b1) `2 )
proof end;

theorem Th27: :: JORDAN22:27
for b1 being Simple_closed_curve
for b2, b3 being Nat holds
( b2 <= b3 implies (UMP (L~ (Cage b1,b3))) `2 <= (UMP (L~ (Cage b1,b2))) `2 )
proof end;

theorem Th28: :: JORDAN22:28
for b1 being Simple_closed_curve
for b2, b3 being Nat holds
( b2 <= b3 implies (LMP (L~ (Cage b1,b2))) `2 <= (LMP (L~ (Cage b1,b3))) `2 )
proof end;

theorem Th29: :: JORDAN22:29
for b1 being Simple_closed_curve
for b2, b3 being Nat holds
( 0 < b2 & b2 <= b3 implies (UMP (Upper_Arc (L~ (Cage b1,b3)))) `2 <= (UMP (Upper_Arc (L~ (Cage b1,b2)))) `2 )
proof end;

theorem Th30: :: JORDAN22:30
for b1 being Simple_closed_curve
for b2, b3 being Nat holds
( 0 < b2 & b2 <= b3 implies (LMP (Lower_Arc (L~ (Cage b1,b2)))) `2 <= (LMP (Lower_Arc (L~ (Cage b1,b3)))) `2 )
proof end;

theorem Th31: :: JORDAN22:31
for b1 being Simple_closed_curve holds W-min b1 in North_Arc b1
proof end;

theorem Th32: :: JORDAN22:32
for b1 being Simple_closed_curve holds E-max b1 in North_Arc b1
proof end;

theorem Th33: :: JORDAN22:33
for b1 being Simple_closed_curve holds W-min b1 in South_Arc b1
proof end;

theorem Th34: :: JORDAN22:34
for b1 being Simple_closed_curve holds E-max b1 in South_Arc b1
proof end;

theorem Th35: :: JORDAN22:35
for b1 being Simple_closed_curve holds UMP b1 in North_Arc b1
proof end;

theorem Th36: :: JORDAN22:36
for b1 being Simple_closed_curve holds LMP b1 in South_Arc b1
proof end;

theorem Th37: :: JORDAN22:37
for b1 being Simple_closed_curve holds North_Arc b1 c= b1
proof end;

theorem Th38: :: JORDAN22:38
for b1 being Simple_closed_curve holds South_Arc b1 c= b1
proof end;

theorem Th39: :: JORDAN22:39
for b1 being Simple_closed_curve holds
( ( LMP b1 in Lower_Arc b1 & UMP b1 in Upper_Arc b1 ) or ( UMP b1 in Lower_Arc b1 & LMP b1 in Upper_Arc b1 ) )
proof end;

theorem Th40: :: JORDAN22:40
for b1 being Simple_closed_curve holds W-bound b1 = W-bound (North_Arc b1)
proof end;

theorem Th41: :: JORDAN22:41
for b1 being Simple_closed_curve holds E-bound b1 = E-bound (North_Arc b1)
proof end;

theorem Th42: :: JORDAN22:42
for b1 being Simple_closed_curve holds W-bound b1 = W-bound (South_Arc b1)
proof end;

theorem Th43: :: JORDAN22:43
for b1 being Simple_closed_curve holds E-bound b1 = E-bound (South_Arc b1)
proof end;