:: JORDAN11 semantic presentation

Lemma1: for b1, b2, b3 being Nat
for b4 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 > 0 & 1 <= b2 & b2 <= width (Gauge b4,b1) & b3 <= b2 & 1 <= b3 & b3 <= width (Gauge b4,b1) & (LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),b2),((Gauge b4,b1) * (Center (Gauge b4,b1)),b3)) /\ (Upper_Arc (L~ (Cage b4,b1))) = {((Gauge b4,b1) * (Center (Gauge b4,b1)),b2)} & (LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),b2),((Gauge b4,b1) * (Center (Gauge b4,b1)),b3)) /\ (Lower_Arc (L~ (Cage b4,b1))) = {((Gauge b4,b1) * (Center (Gauge b4,b1)),b3)} implies LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),b2),((Gauge b4,b1) * (Center (Gauge b4,b1)),b3) c= Cl (RightComp (Cage b4,b1)) )
proof end;

Lemma2: for b1 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
ex b2 being Nat st b2 is_sufficiently_large_for b1
proof end;

definition
let c1 be being_simple_closed_curve Subset of (TOP-REAL 2);
func ApproxIndex c1 -> Nat means :Def1: :: JORDAN11:def 1
( a2 is_sufficiently_large_for a1 & ( for b1 being Nat holds
( b1 is_sufficiently_large_for a1 implies b1 >= a2 ) ) );
existence
ex b1 being Nat st
( b1 is_sufficiently_large_for c1 & ( for b2 being Nat holds
( b2 is_sufficiently_large_for c1 implies b2 >= b1 ) ) )
proof end;
uniqueness
for b1, b2 being Nat holds
( b1 is_sufficiently_large_for c1 & ( for b3 being Nat holds
( b3 is_sufficiently_large_for c1 implies b3 >= b1 ) ) & b2 is_sufficiently_large_for c1 & ( for b3 being Nat holds
( b3 is_sufficiently_large_for c1 implies b3 >= b2 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def1 defines ApproxIndex JORDAN11:def 1 :
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2 being Nat holds
( b2 = ApproxIndex b1 iff ( b2 is_sufficiently_large_for b1 & ( for b3 being Nat holds
( b3 is_sufficiently_large_for b1 implies b3 >= b2 ) ) ) );

theorem Th1: :: JORDAN11:1
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2) holds ApproxIndex b1 >= 1
proof end;

definition
let c1 be being_simple_closed_curve Subset of (TOP-REAL 2);
func Y-InitStart c1 -> Nat means :Def2: :: JORDAN11:def 2
( a2 < width (Gauge a1,(ApproxIndex a1)) & cell (Gauge a1,(ApproxIndex a1)),((X-SpanStart a1,(ApproxIndex a1)) -' 1),a2 c= BDD a1 & ( for b1 being Nat holds
( b1 < width (Gauge a1,(ApproxIndex a1)) & cell (Gauge a1,(ApproxIndex a1)),((X-SpanStart a1,(ApproxIndex a1)) -' 1),b1 c= BDD a1 implies b1 >= a2 ) ) );
existence
ex b1 being Nat st
( b1 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b1 c= BDD c1 & ( for b2 being Nat holds
( b2 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b2 c= BDD c1 implies b2 >= b1 ) ) )
proof end;
uniqueness
for b1, b2 being Nat holds
( b1 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b1 c= BDD c1 & ( for b3 being Nat holds
( b3 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b3 c= BDD c1 implies b3 >= b1 ) ) & b2 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b2 c= BDD c1 & ( for b3 being Nat holds
( b3 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b3 c= BDD c1 implies b3 >= b2 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines Y-InitStart JORDAN11:def 2 :
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2 being Nat holds
( b2 = Y-InitStart b1 iff ( b2 < width (Gauge b1,(ApproxIndex b1)) & cell (Gauge b1,(ApproxIndex b1)),((X-SpanStart b1,(ApproxIndex b1)) -' 1),b2 c= BDD b1 & ( for b3 being Nat holds
( b3 < width (Gauge b1,(ApproxIndex b1)) & cell (Gauge b1,(ApproxIndex b1)),((X-SpanStart b1,(ApproxIndex b1)) -' 1),b3 c= BDD b1 implies b3 >= b2 ) ) ) );

theorem Th2: :: JORDAN11:2
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
Y-InitStart b1 > 1
proof end;

theorem Th3: :: JORDAN11:3
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
(Y-InitStart b1) + 1 < width (Gauge b1,(ApproxIndex b1))
proof end;

definition
let c1 be being_simple_closed_curve Subset of (TOP-REAL 2);
let c2 be Nat;
assume E8: c2 is_sufficiently_large_for c1 ;
E9: c2 >= ApproxIndex c1 by E8, Def1;
set c3 = X-SpanStart c1,c2;
func Y-SpanStart c1,c2 -> Nat means :Def3: :: JORDAN11:def 3
( a3 <= width (Gauge a1,a2) & ( for b1 being Nat holds
( a3 <= b1 & b1 <= ((2 |^ (a2 -' (ApproxIndex a1))) * ((Y-InitStart a1) -' 2)) + 2 implies cell (Gauge a1,a2),((X-SpanStart a1,a2) -' 1),b1 c= BDD a1 ) ) & ( for b1 being Nat holds
( b1 <= width (Gauge a1,a2) & ( for b2 being Nat holds
( b1 <= b2 & b2 <= ((2 |^ (a2 -' (ApproxIndex a1))) * ((Y-InitStart a1) -' 2)) + 2 implies cell (Gauge a1,a2),((X-SpanStart a1,a2) -' 1),b2 c= BDD a1 ) ) implies b1 >= a3 ) ) );
existence
ex b1 being Nat st
( b1 <= width (Gauge c1,c2) & ( for b2 being Nat holds
( b1 <= b2 & b2 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b2 c= BDD c1 ) ) & ( for b2 being Nat holds
( b2 <= width (Gauge c1,c2) & ( for b3 being Nat holds
( b2 <= b3 & b3 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b3 c= BDD c1 ) ) implies b2 >= b1 ) ) )
proof end;
uniqueness
for b1, b2 being Nat holds
( b1 <= width (Gauge c1,c2) & ( for b3 being Nat holds
( b1 <= b3 & b3 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b3 c= BDD c1 ) ) & ( for b3 being Nat holds
( b3 <= width (Gauge c1,c2) & ( for b4 being Nat holds
( b3 <= b4 & b4 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b4 c= BDD c1 ) ) implies b3 >= b1 ) ) & b2 <= width (Gauge c1,c2) & ( for b3 being Nat holds
( b2 <= b3 & b3 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b3 c= BDD c1 ) ) & ( for b3 being Nat holds
( b3 <= width (Gauge c1,c2) & ( for b4 being Nat holds
( b3 <= b4 & b4 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b4 c= BDD c1 ) ) implies b3 >= b2 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines Y-SpanStart JORDAN11:def 3 :
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2 being Nat holds
( b2 is_sufficiently_large_for b1 implies for b3 being Nat holds
( b3 = Y-SpanStart b1,b2 iff ( b3 <= width (Gauge b1,b2) & ( for b4 being Nat holds
( b3 <= b4 & b4 <= ((2 |^ (b2 -' (ApproxIndex b1))) * ((Y-InitStart b1) -' 2)) + 2 implies cell (Gauge b1,b2),((X-SpanStart b1,b2) -' 1),b4 c= BDD b1 ) ) & ( for b4 being Nat holds
( b4 <= width (Gauge b1,b2) & ( for b5 being Nat holds
( b4 <= b5 & b5 <= ((2 |^ (b2 -' (ApproxIndex b1))) * ((Y-InitStart b1) -' 2)) + 2 implies cell (Gauge b1,b2),((X-SpanStart b1,b2) -' 1),b5 c= BDD b1 ) ) implies b4 >= b3 ) ) ) ) );

theorem Th4: :: JORDAN11:4
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 is_sufficiently_large_for b2 implies X-SpanStart b2,b1 = ((2 |^ (b1 -' (ApproxIndex b2))) * ((X-SpanStart b2,(ApproxIndex b2)) - 2)) + 2 )
proof end;

theorem Th5: :: JORDAN11:5
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 is_sufficiently_large_for b2 implies Y-SpanStart b2,b1 <= ((2 |^ (b1 -' (ApproxIndex b2))) * ((Y-InitStart b2) -' 2)) + 2 )
proof end;

theorem Th6: :: JORDAN11:6
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 is_sufficiently_large_for b2 implies cell (Gauge b2,b1),((X-SpanStart b2,b1) -' 1),(Y-SpanStart b2,b1) c= BDD b2 )
proof end;

theorem Th7: :: JORDAN11:7
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 is_sufficiently_large_for b2 implies ( 1 < Y-SpanStart b2,b1 & Y-SpanStart b2,b1 <= width (Gauge b2,b1) ) )
proof end;

theorem Th8: :: JORDAN11:8
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 is_sufficiently_large_for b2 implies [(X-SpanStart b2,b1),(Y-SpanStart b2,b1)] in Indices (Gauge b2,b1) )
proof end;

theorem Th9: :: JORDAN11:9
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 is_sufficiently_large_for b2 implies [((X-SpanStart b2,b1) -' 1),(Y-SpanStart b2,b1)] in Indices (Gauge b2,b1) )
proof end;

theorem Th10: :: JORDAN11:10
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
not ( b1 is_sufficiently_large_for b2 & not cell (Gauge b2,b1),((X-SpanStart b2,b1) -' 1),((Y-SpanStart b2,b1) -' 1) meets b2 )
proof end;

theorem Th11: :: JORDAN11:11
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 is_sufficiently_large_for b2 implies cell (Gauge b2,b1),((X-SpanStart b2,b1) -' 1),(Y-SpanStart b2,b1) misses b2 )
proof end;

theorem Th12: :: JORDAN11:12
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2 being Simple_closed_curve holds
UBD b1 meets UBD b2
proof end;

theorem Th13: :: JORDAN11:13
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
not ( b2 in UBD b1 & b3 in BDD b1 & not dist b2,b1 < dist b2,b3 )
proof end;

registration
let c1 be being_simple_closed_curve Subset of (TOP-REAL 2);
cluster BDD a1 -> non empty ;
coherence
not BDD c1 is empty
proof end;
end;

theorem Th14: :: JORDAN11:14
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( not b2 in BDD b1 implies dist b2,b1 <= dist b2,(BDD b1) )
proof end;

theorem Th15: :: JORDAN11:15
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2 being Simple_closed_curve holds
not ( b1 c= BDD b2 & b2 c= BDD b1 )
proof end;

theorem Th16: :: JORDAN11:16
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2)
for b2 being Simple_closed_curve holds
( b1 c= BDD b2 implies b2 c= UBD b1 )
proof end;

theorem Th17: :: JORDAN11:17
for b1 being Nat
for b2 being being_simple_closed_curve Subset of (TOP-REAL 2) holds L~ (Cage b2,b1) c= UBD b2
proof end;