:: JGRAPH_8 semantic presentation

Lemma1: I[01] = TopSpaceMetr (Closed-Interval-MSpace 0,1)
by TOPMETR:27, TOPMETR:def 8;

Lemma2: for b1 being set
for b2 being FinSequence holds
( 1 <= len b2 implies ( (b2 ^ <*b1*>) . 1 = b2 . 1 & (<*b1*> ^ b2) . ((len b2) + 1) = b2 . (len b2) ) )
proof end;

Lemma3: for b1 being FinSequence of REAL holds
( ( for b2 being Nat holds
not ( 1 <= b2 & b2 < len b1 & not b1 /. b2 < b1 /. (b2 + 1) ) ) implies b1 is increasing )
proof end;

registration
let c1, c2, c3, c4 be real number ;
cluster closed_inside_of_rectangle a1,a2,a3,a4 -> convex ;
coherence
closed_inside_of_rectangle c1,c2,c3,c4 is convex
proof end;
end;

registration
let c1, c2, c3, c4 be real number ;
cluster Trectangle a1,a2,a3,a4 -> convex ;
coherence
Trectangle c1,c2,c3,c4 is convex
proof end;
end;

theorem Th1: :: JGRAPH_8:1
for b1 being Nat
for b2 being positive real number
for b3 being continuous Function of I[01] ,(TOP-REAL b1) holds
ex b4 being FinSequence of REAL st
( b4 . 1 = 0 & b4 . (len b4) = 1 & 5 <= len b4 & rng b4 c= the carrier of I[01] & b4 is increasing & ( for b5 being Nat
for b6 being Subset of I[01]
for b7 being Subset of (Euclid b1) holds
not ( 1 <= b5 & b5 < len b4 & b6 = [.(b4 /. b5),(b4 /. (b5 + 1)).] & b7 = b3 .: b6 & not diameter b7 < b2 ) ) )
proof end;

theorem Th2: :: JGRAPH_8:2
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1)
for b4 being Subset of (TOP-REAL b1) holds
( b4 c= LSeg b2,b3 & b2 in b4 & b3 in b4 & b4 is connected implies b4 = LSeg b2,b3 )
proof end;

theorem Th3: :: JGRAPH_8:3
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1)
for b4 being Path of b2,b3 holds
( rng b4 c= LSeg b2,b3 implies rng b4 = LSeg b2,b3 )
proof end;

theorem Th4: :: JGRAPH_8:4
for b1, b2 being non empty Subset of (TOP-REAL 2)
for b3, b4, b5, b6 being Point of (TOP-REAL 2)
for b7 being Path of b3,b4
for b8 being Path of b5,b6 holds
not ( rng b7 = b1 & rng b8 = b2 & ( for b9 being Point of (TOP-REAL 2) holds
( b9 in b1 implies ( b3 `1 <= b9 `1 & b9 `1 <= b4 `1 ) ) ) & ( for b9 being Point of (TOP-REAL 2) holds
( b9 in b2 implies ( b3 `1 <= b9 `1 & b9 `1 <= b4 `1 ) ) ) & ( for b9 being Point of (TOP-REAL 2) holds
( b9 in b1 implies ( b5 `2 <= b9 `2 & b9 `2 <= b6 `2 ) ) ) & ( for b9 being Point of (TOP-REAL 2) holds
( b9 in b2 implies ( b5 `2 <= b9 `2 & b9 `2 <= b6 `2 ) ) ) & not b1 meets b2 )
proof end;

theorem Th5: :: JGRAPH_8:5
for b1, b2, b3, b4 being real number
for b5, b6 being continuous Function of I[01] ,(TOP-REAL 2)
for b7, b8 being Point of I[01] holds
not ( b7 = 0 & b8 = 1 & (b5 . b7) `1 = b1 & (b5 . b8) `1 = b2 & (b6 . b7) `2 = b3 & (b6 . b8) `2 = b4 & ( for b9 being Point of I[01] holds
( b1 <= (b5 . b9) `1 & (b5 . b9) `1 <= b2 & b1 <= (b6 . b9) `1 & (b6 . b9) `1 <= b2 & b3 <= (b5 . b9) `2 & (b5 . b9) `2 <= b4 & b3 <= (b6 . b9) `2 & (b6 . b9) `2 <= b4 ) ) & not rng b5 meets rng b6 )
proof end;

theorem Th6: :: JGRAPH_8:6
for b1, b2, b3, b4 being real number
for b5, b6, b7, b8 being Point of (Trectangle b1,b2,b3,b4)
for b9 being Path of b5,b6
for b10 being Path of b8,b7
for b11, b12, b13, b14 being Point of (TOP-REAL 2) holds
not ( b11 `1 = b1 & b12 `1 = b2 & b13 `2 = b3 & b14 `2 = b4 & b5 = b11 & b6 = b12 & b7 = b13 & b8 = b14 & ( for b15, b16 being Point of I[01] holds
not b9 . b15 = b10 . b16 ) )
proof end;