:: JORDAN5B semantic presentation

theorem Th1: :: JORDAN5B:1
for b1 being Nat holds
not ( 1 <= b1 & not b1 -' 1 < b1 )
proof end;

theorem Th2: :: JORDAN5B:2
for b1, b2 being Nat holds
( b1 + 1 <= b2 implies 1 <= b2 -' b1 )
proof end;

theorem Th3: :: JORDAN5B:3
for b1, b2 being Nat holds
( 1 <= b1 & 1 <= b2 implies (b2 -' b1) + 1 <= b2 )
proof end;

Lemma2: for b1 being real number holds
( 0 <= b1 & b1 <= 1 implies ( 0 <= 1 - b1 & 1 - b1 <= 1 ) )
proof end;

theorem Th4: :: JORDAN5B:4
for b1 being real number holds
( b1 in the carrier of I[01] implies 1 - b1 in the carrier of I[01] )
proof end;

theorem Th5: :: JORDAN5B:5
for b1, b2, b3 being Point of (TOP-REAL 2) holds
( b1 `2 <> b2 `2 & b3 in LSeg b1,b2 & b3 `2 = b1 `2 implies b3 = b1 )
proof end;

theorem Th6: :: JORDAN5B:6
for b1, b2, b3 being Point of (TOP-REAL 2) holds
( b1 `1 <> b2 `1 & b3 in LSeg b1,b2 & b3 `1 = b1 `1 implies b3 = b1 )
proof end;

Lemma3: for b1 being Point of I[01] holds
b1 is Real
proof end;

theorem Th7: :: JORDAN5B:7
for b1 being FinSequence of (TOP-REAL 2)
for b2 being non empty Subset of (TOP-REAL 2)
for b3 being Function of I[01] ,((TOP-REAL 2) | b2)
for b4 being Nat holds
not ( 1 <= b4 & b4 + 1 <= len b1 & b1 is_S-Seq & b2 = L~ b1 & b3 is_homeomorphism & b3 . 0 = b1 /. 1 & b3 . 1 = b1 /. (len b1) & ( for b5, b6 being Real holds
not ( b5 < b6 & 0 <= b5 & b5 <= 1 & 0 <= b6 & b6 <= 1 & LSeg b1,b4 = b3 .: [.b5,b6.] & b3 . b5 = b1 /. b4 & b3 . b6 = b1 /. (b4 + 1) ) ) )
proof end;

theorem Th8: :: JORDAN5B:8
for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being non empty Subset of (TOP-REAL 2)
for b4 being Function of I[01] ,((TOP-REAL 2) | b2)
for b5 being Nat
for b6 being non empty Subset of I[01] holds
not ( b1 is_S-Seq & b4 is_homeomorphism & b4 . 0 = b1 /. 1 & b4 . 1 = b1 /. (len b1) & 1 <= b5 & b5 + 1 <= len b1 & b4 .: b6 = LSeg b1,b5 & b2 = L~ b1 & b3 = LSeg b1,b5 & ( for b7 being Function of (I[01] | b6),((TOP-REAL 2) | b3) holds
not ( b7 = b4 | b6 & b7 is_homeomorphism ) ) )
proof end;

theorem Th9: :: JORDAN5B:9
for b1, b2, b3 being Point of (TOP-REAL 2) holds
( b1 <> b2 & b3 in LSeg b1,b2 implies LE b3,b3,b1,b2 )
proof end;

theorem Th10: :: JORDAN5B:10
for b1, b2, b3 being Point of (TOP-REAL 2) holds
( b2 <> b3 & b1 in LSeg b2,b3 implies LE b2,b1,b2,b3 )
proof end;

theorem Th11: :: JORDAN5B:11
for b1, b2, b3 being Point of (TOP-REAL 2) holds
( b1 in LSeg b2,b3 & b2 <> b3 implies LE b1,b3,b2,b3 )
proof end;

theorem Th12: :: JORDAN5B:12
for b1, b2, b3, b4, b5 being Point of (TOP-REAL 2) holds
( b1 <> b2 & LE b3,b4,b1,b2 & LE b4,b5,b1,b2 implies LE b3,b5,b1,b2 )
proof end;

theorem Th13: :: JORDAN5B:13
for b1, b2 being Point of (TOP-REAL 2) holds
( b1 <> b2 implies LSeg b1,b2 = { b3 where B is Point of (TOP-REAL 2) : ( LE b1,b3,b1,b2 & LE b3,b2,b1,b2 ) } )
proof end;

theorem Th14: :: JORDAN5B:14
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1) holds
( b2 is_an_arc_of b3,b4 implies b2 is_an_arc_of b4,b3 )
proof end;

theorem Th15: :: JORDAN5B:15
for b1 being Nat
for b2 being FinSequence of (TOP-REAL 2)
for b3 being Subset of (TOP-REAL 2) holds
( b2 is_S-Seq & 1 <= b1 & b1 + 1 <= len b2 & b3 = LSeg b2,b1 implies b3 is_an_arc_of b2 /. b1,b2 /. (b1 + 1) )
proof end;

theorem Th16: :: JORDAN5B:16
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Nat holds
( 1 <= b2 & b2 <= len b1 & b1 is_S-Seq & b1 /. 1 in L~ (mid b1,b2,(len b1)) implies b2 = 1 )
proof end;

theorem Th17: :: JORDAN5B:17
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b1 is_S-Seq & b2 = b1 . (len b1) implies L_Cut b1,b2 = <*b2*> )
proof end;

theorem Th18: :: JORDAN5B:18
canceled;

theorem Th19: :: JORDAN5B:19
canceled;

theorem Th20: :: JORDAN5B:20
canceled;

theorem Th21: :: JORDAN5B:21
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b2 <> b1 . (len b1) & b1 is_S-Seq implies Index b2,(L_Cut b1,b2) = 1 )
proof end;

theorem Th22: :: JORDAN5B:22
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b1 is_S-Seq & b2 <> b1 . (len b1) implies b2 in L~ (L_Cut b1,b2) )
proof end;

theorem Th23: :: JORDAN5B:23
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b1 is_S-Seq & b2 <> b1 . 1 implies b2 in L~ (R_Cut b1,b2) )
proof end;

theorem Th24: :: JORDAN5B:24
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b1 is one-to-one implies B_Cut b1,b2,b2 = <*b2*> )
proof end;

Lemma10: for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
not ( b2 in L~ b1 & b3 in L~ b1 & b2 <> b1 . (len b1) & b3 <> b1 . (len b1) & b1 is_S-Seq & not b2 in L~ (L_Cut b1,b3) & not b3 in L~ (L_Cut b1,b2) )
proof end;

theorem Th25: :: JORDAN5B:25
for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b3 in L~ b1 & b3 <> b1 . (len b1) & b2 = b1 . (len b1) & b1 is_S-Seq implies b2 in L~ (L_Cut b1,b3) )
proof end;

theorem Th26: :: JORDAN5B:26
for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
not ( b2 <> b1 . (len b1) & b2 in L~ b1 & b3 in L~ b1 & b1 is_S-Seq & not b2 in L~ (L_Cut b1,b3) & not b3 in L~ (L_Cut b1,b2) )
proof end;

Lemma12: for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b3 in L~ b1 & not ( not Index b2,b1 < Index b3,b1 & not ( Index b2,b1 = Index b3,b1 & LE b2,b3,b1 /. (Index b2,b1),b1 /. ((Index b2,b1) + 1) ) ) & b2 <> b3 implies L~ (B_Cut b1,b2,b3) c= L~ b1 )
proof end;

theorem Th27: :: JORDAN5B:27
for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b3 in L~ b1 & b1 is_S-Seq implies L~ (B_Cut b1,b2,b3) c= L~ b1 )
proof end;

theorem Th28: :: JORDAN5B:28
for b1 being non constant standard special_circular_sequence
for b2, b3 being Nat holds
( 1 <= b2 & b3 <= len (GoB b1) & b2 < b3 implies (LSeg ((GoB b1) * 1,(width (GoB b1))),((GoB b1) * b2,(width (GoB b1)))) /\ (LSeg ((GoB b1) * b3,(width (GoB b1))),((GoB b1) * (len (GoB b1)),(width (GoB b1)))) = {} )
proof end;

theorem Th29: :: JORDAN5B:29
for b1 being non constant standard special_circular_sequence
for b2, b3 being Nat holds
( 1 <= b2 & b3 <= width (GoB b1) & b2 < b3 implies (LSeg ((GoB b1) * (len (GoB b1)),1),((GoB b1) * (len (GoB b1)),b2)) /\ (LSeg ((GoB b1) * (len (GoB b1)),b3),((GoB b1) * (len (GoB b1)),(width (GoB b1)))) = {} )
proof end;

theorem Th30: :: JORDAN5B:30
for b1 being FinSequence of (TOP-REAL 2) holds
( b1 is_S-Seq implies L_Cut b1,(b1 /. 1) = b1 )
proof end;

theorem Th31: :: JORDAN5B:31
for b1 being FinSequence of (TOP-REAL 2) holds
( b1 is_S-Seq implies R_Cut b1,(b1 /. (len b1)) = b1 )
proof end;

theorem Th32: :: JORDAN5B:32
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 implies b2 in LSeg (b1 /. (Index b2,b1)),(b1 /. ((Index b2,b1) + 1)) )
proof end;

theorem Th33: :: JORDAN5B:33
for b1 being FinSequence of (TOP-REAL 2)
for b2 being Nat holds
( b1 is unfolded & b1 is s.n.c. & b1 is one-to-one & len b1 >= 2 & b1 /. 1 in LSeg b1,b2 implies b2 = 1 )
proof end;

theorem Th34: :: JORDAN5B:34
for b1 being non constant standard special_circular_sequence
for b2 being Nat
for b3 being Subset of (TOP-REAL 2) holds
( 1 <= b2 & b2 <= width (GoB b1) & b3 = LSeg ((GoB b1) * 1,b2),((GoB b1) * (len (GoB b1)),b2) implies b3 is_S-P_arc_joining (GoB b1) * 1,b2,(GoB b1) * (len (GoB b1)),b2 )
proof end;

theorem Th35: :: JORDAN5B:35
for b1 being non constant standard special_circular_sequence
for b2 being Nat
for b3 being Subset of (TOP-REAL 2) holds
( 1 <= b2 & b2 <= len (GoB b1) & b3 = LSeg ((GoB b1) * b2,1),((GoB b1) * b2,(width (GoB b1))) implies b3 is_S-P_arc_joining (GoB b1) * b2,1,(GoB b1) * b2,(width (GoB b1)) )
proof end;

theorem Th36: :: JORDAN5B:36
for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b3 in L~ b1 & not ( not Index b2,b1 < Index b3,b1 & not ( Index b2,b1 = Index b3,b1 & LE b2,b3,b1 /. (Index b2,b1),b1 /. ((Index b2,b1) + 1) ) ) & b2 <> b3 implies L~ (B_Cut b1,b2,b3) c= L~ b1 ) by Lemma12;