:: JORDAN16 semantic presentation

theorem Th1: :: JORDAN16:1
for b1 being non empty finite Subset of REAL
for b2 being Real holds
not ( ( for b3 being Real holds
not ( b3 in b1 & not b3 < b2 ) ) & not max b1 < b2 )
proof end;

registration
let c1 be Nat;
cluster trivial Element of K22(the carrier of (TOP-REAL a1));
existence
ex b1 being Subset of (TOP-REAL c1) st b1 is trivial
proof end;
end;

theorem Th2: :: JORDAN16:2
for b1, b2, b3, b4 being set holds
( b1 in b4 & b2 in b4 & b3 in b4 implies {b1,b2,b3} c= b4 )
proof end;

theorem Th3: :: JORDAN16:3
for b1 being Nat holds {} (TOP-REAL b1) is Bounded by JORDAN2C:73;

theorem Th4: :: JORDAN16:4
for b1 being Simple_closed_curve holds
Lower_Arc b1 <> Upper_Arc b1
proof end;

theorem Th5: :: JORDAN16:5
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4, b5 being Point of (TOP-REAL 2) holds Segment b1,b2,b3,b4,b5 c= b1
proof end;

theorem Th6: :: JORDAN16:6
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2 c= b3 implies b1 | b2 is SubSpace of b1 | b3 ) by TOPMETR:29;

theorem Th7: :: JORDAN16:7
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4 being Point of (TOP-REAL 2) holds
( b1 is_an_arc_of b2,b3 & b4 in b1 implies b4 in L_Segment b1,b2,b3,b4 )
proof end;

theorem Th8: :: JORDAN16:8
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4 being Point of (TOP-REAL 2) holds
( b1 is_an_arc_of b2,b3 & b4 in b1 implies b4 in R_Segment b1,b2,b3,b4 )
proof end;

theorem Th9: :: JORDAN16:9
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4, b5 being Point of (TOP-REAL 2) holds
( b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 implies ( b4 in Segment b1,b2,b3,b4,b5 & b5 in Segment b1,b2,b3,b4,b5 ) )
proof end;

theorem Th10: :: JORDAN16:10
for b1 being Simple_closed_curve
for b2, b3 being Point of (TOP-REAL 2) holds Segment b2,b3,b1 c= b1
proof end;

theorem Th11: :: JORDAN16:11
for b1 being Simple_closed_curve
for b2, b3 being Point of (TOP-REAL 2) holds
not ( b2 in b1 & b3 in b1 & not LE b2,b3,b1 & not LE b3,b2,b1 )
proof end;

theorem Th12: :: JORDAN16:12
for b1, b2 being non empty TopSpace
for b3 being non empty SubSpace of b2
for b4 being Function of b1,b2
for b5 being Function of b1,b3 holds
( b4 = b5 & b4 is continuous implies b5 is continuous )
proof end;

theorem Th13: :: JORDAN16:13
for b1, b2 being non empty TopSpace
for b3 being non empty SubSpace of b1
for b4 being non empty SubSpace of b2
for b5 being Function of b1,b2 holds
( b5 is_homeomorphism implies for b6 being Function of b3,b4 holds
( b6 = b5 | b3 & b6 is onto implies b6 is_homeomorphism ) )
proof end;

theorem Th14: :: JORDAN16:14
for b1, b2, b3 being Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
not ( b1 is_an_arc_of b4,b5 & b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 & b2 /\ b3 = {b4,b5} & b1 c= b2 \/ b3 & not b1 = b2 & not b1 = b3 )
proof end;

theorem Th15: :: JORDAN16:15
for b1 being Simple_closed_curve
for b2, b3 being Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
( b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 & b2 c= b1 & b3 c= b1 & b2 <> b3 implies ( b2 \/ b3 = b1 & b2 /\ b3 = {b4,b5} ) )
proof end;

theorem Th16: :: JORDAN16:16
for b1, b2 being Subset of (TOP-REAL 2)
for b3, b4, b5, b6 being Point of (TOP-REAL 2) holds
not ( b1 is_an_arc_of b3,b4 & b1 /\ b2 = {b5,b6} & not b1 <> b2 )
proof end;

theorem Th17: :: JORDAN16:17
for b1 being Simple_closed_curve
for b2, b3 being Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
( b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 & b2 c= b1 & b3 c= b1 & b2 /\ b3 = {b4,b5} implies b2 \/ b3 = b1 )
proof end;

theorem Th18: :: JORDAN16:18
for b1 being Simple_closed_curve
for b2, b3 being Subset of (TOP-REAL 2)
for b4, b5 being Point of (TOP-REAL 2) holds
( b2 c= b1 & b3 c= b1 & b2 <> b3 & b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 implies for b6 being Subset of (TOP-REAL 2) holds
not ( b6 is_an_arc_of b4,b5 & b6 c= b1 & not b6 = b2 & not b6 = b3 ) )
proof end;

theorem Th19: :: JORDAN16:19
for b1 being Simple_closed_curve
for b2 being non empty Subset of (TOP-REAL 2) holds
not ( b2 is_an_arc_of W-min b1, E-max b1 & b2 c= b1 & not b2 = Lower_Arc b1 & not b2 = Upper_Arc b1 )
proof end;

theorem Th20: :: JORDAN16:20
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4, b5 being Point of (TOP-REAL 2) holds
not ( b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & ( for b6 being Function of I[01] ,((TOP-REAL 2) | b1)
for b7, b8 being Real holds
not ( b6 is_homeomorphism & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & b6 . b8 = b5 & 0 <= b7 & b7 <= b8 & b8 <= 1 ) ) )
proof end;

theorem Th21: :: JORDAN16:21
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4, b5 being Point of (TOP-REAL 2) holds
not ( b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & b4 <> b5 & ( for b6 being Function of I[01] ,((TOP-REAL 2) | b1)
for b7, b8 being Real holds
not ( b6 is_homeomorphism & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & b6 . b8 = b5 & 0 <= b7 & b7 < b8 & b8 <= 1 ) ) )
proof end;

theorem Th22: :: JORDAN16:22
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4, b5 being Point of (TOP-REAL 2) holds
not ( b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & Segment b1,b2,b3,b4,b5 is empty )
proof end;

theorem Th23: :: JORDAN16:23
for b1 being Simple_closed_curve
for b2 being Point of (TOP-REAL 2) holds
( b2 in b1 implies ( b2 in Segment b2,(W-min b1),b1 & W-min b1 in Segment b2,(W-min b1),b1 ) )
proof end;

definition
let c1 be PartFunc of REAL , REAL ;
attr a1 is continuous means :Def1: :: JORDAN16:def 1
a1 is_continuous_on dom a1;
end;

:: deftheorem Def1 defines continuous JORDAN16:def 1 :
for b1 being PartFunc of REAL , REAL holds
( b1 is continuous iff b1 is_continuous_on dom b1 );

definition
let c1 be Function of REAL , REAL ;
redefine attr a1 is continuous means :: JORDAN16:def 2
a1 is_continuous_on REAL ;
compatibility
( c1 is continuous iff c1 is_continuous_on REAL )
proof end;
end;

:: deftheorem Def2 defines continuous JORDAN16:def 2 :
for b1 being Function of REAL , REAL holds
( b1 is continuous iff b1 is_continuous_on REAL );

definition
let c1, c2 be real number ;
func AffineMap c1,c2 -> Function of REAL , REAL means :Def3: :: JORDAN16:def 3
for b1 being real number holds a3 . b1 = (a1 * b1) + a2;
existence
ex b1 being Function of REAL , REAL st
for b2 being real number holds b1 . b2 = (c1 * b2) + c2
proof end;
uniqueness
for b1, b2 being Function of REAL , REAL holds
( ( for b3 being real number holds b1 . b3 = (c1 * b3) + c2 ) & ( for b3 being real number holds b2 . b3 = (c1 * b3) + c2 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines AffineMap JORDAN16:def 3 :
for b1, b2 being real number
for b3 being Function of REAL , REAL holds
( b3 = AffineMap b1,b2 iff for b4 being real number holds b3 . b4 = (b1 * b4) + b2 );

registration
let c1, c2 be real number ;
cluster AffineMap a1,a2 -> continuous ;
coherence
AffineMap c1,c2 is continuous
proof end;
end;

registration
cluster continuous Relation of REAL , REAL ;
existence
ex b1 being Function of REAL , REAL st b1 is continuous
proof end;
end;

theorem Th24: :: JORDAN16:24
for b1, b2 being continuous PartFunc of REAL , REAL holds
b2 * b1 is continuous PartFunc of REAL , REAL
proof end;

theorem Th25: :: JORDAN16:25
for b1, b2 being real number holds (AffineMap b1,b2) . 0 = b2
proof end;

theorem Th26: :: JORDAN16:26
for b1, b2 being real number holds (AffineMap b1,b2) . 1 = b1 + b2
proof end;

theorem Th27: :: JORDAN16:27
for b1, b2 being real number holds
( b1 <> 0 implies AffineMap b1,b2 is one-to-one )
proof end;

theorem Th28: :: JORDAN16:28
for b1, b2, b3, b4 being real number holds
not ( b1 > 0 & b3 < b4 & not (AffineMap b1,b2) . b3 < (AffineMap b1,b2) . b4 )
proof end;

theorem Th29: :: JORDAN16:29
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & b3 < b4 & not (AffineMap b1,b2) . b3 > (AffineMap b1,b2) . b4 )
proof end;

theorem Th30: :: JORDAN16:30
for b1, b2, b3, b4 being real number holds
( b1 >= 0 & b3 <= b4 implies (AffineMap b1,b2) . b3 <= (AffineMap b1,b2) . b4 )
proof end;

theorem Th31: :: JORDAN16:31
for b1, b2, b3, b4 being real number holds
( b1 <= 0 & b3 <= b4 implies (AffineMap b1,b2) . b3 >= (AffineMap b1,b2) . b4 )
proof end;

theorem Th32: :: JORDAN16:32
for b1, b2 being real number holds
( b1 <> 0 implies rng (AffineMap b1,b2) = REAL )
proof end;

theorem Th33: :: JORDAN16:33
for b1, b2 being real number holds
( b1 <> 0 implies (AffineMap b1,b2) " = AffineMap (b1 " ),(- (b2 / b1)) )
proof end;

theorem Th34: :: JORDAN16:34
for b1, b2 being real number holds
( b1 > 0 implies (AffineMap b1,b2) .: [.0,1.] = [.b2,(b1 + b2).] )
proof end;

theorem Th35: :: JORDAN16:35
for b1 being Function of R^1 ,R^1
for b2, b3 being Real holds
( b2 <> 0 & b1 = AffineMap b2,b3 implies b1 is_homeomorphism )
proof end;

theorem Th36: :: JORDAN16:36
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4, b5 being Point of (TOP-REAL 2) holds
( b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & b4 <> b5 implies Segment b1,b2,b3,b4,b5 is_an_arc_of b4,b5 )
proof end;

theorem Th37: :: JORDAN16:37
for b1 being Simple_closed_curve
for b2, b3 being Point of (TOP-REAL 2)
for b4 being Subset of (TOP-REAL 2) holds
not ( b4 c= b1 & b4 is_an_arc_of b2,b3 & W-min b1 in b4 & E-max b1 in b4 & not Upper_Arc b1 c= b4 & not Lower_Arc b1 c= b4 )
proof end;

theorem Th38: :: JORDAN16:38
for b1 being Subset of (TOP-REAL 2)
for b2, b3, b4, b5 being Point of (TOP-REAL 2) holds
not ( b1 is_an_arc_of b2,b3 & b4 in b1 & b5 in b1 & b4 <> b2 & b4 <> b3 & b5 <> b2 & b5 <> b3 & b4 <> b5 & ( for b6 being non empty Subset of (TOP-REAL 2) holds
not ( b6 is_an_arc_of b4,b5 & b6 c= b1 & b6 misses {b2,b3} ) ) )
proof end;

theorem Th39: :: JORDAN16:39
for b1 being non empty Subset of (TOP-REAL 2)
for b2, b3, b4 being Point of (TOP-REAL 2) holds
( b1 is_an_arc_of b2,b3 & b4 in b1 & b2 <> b4 implies Segment b1,b2,b3,b2,b4 is_an_arc_of b2,b4 )
proof end;