:: JORDAN24 semantic presentation

definition
let c1 be Nat;
let c2 be Subset of (TOP-REAL c1);
let c3, c4 be Point of (TOP-REAL c1);
pred c3,c4 realize-max-dist-in c2 means :Def1: :: JORDAN24:def 1
( a3 in a2 & a4 in a2 & ( for b1, b2 being Point of (TOP-REAL a1) holds
( b1 in a2 & b2 in a2 implies dist a3,a4 >= dist b1,b2 ) ) );
end;

:: deftheorem Def1 defines realize-max-dist-in JORDAN24:def 1 :
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1) holds
( b3,b4 realize-max-dist-in b2 iff ( b3 in b2 & b4 in b2 & ( for b5, b6 being Point of (TOP-REAL b1) holds
( b5 in b2 & b6 in b2 implies dist b3,b4 >= dist b5,b6 ) ) ) );

set c1 = - 1;

set c2 = 1;

set c3 = |[(- 1),0]|;

set c4 = |[1,0]|;

theorem Th1: :: JORDAN24:1
for b1 being Simple_closed_curve holds
ex b2, b3 being Point of (TOP-REAL 2) st b2,b3 realize-max-dist-in b1
proof end;

definition
let c5 be non empty MetrStruct ;
let c6 be Function of (TopSpaceMetr c5),(TopSpaceMetr c5);
attr a2 is isometric means :Def2: :: JORDAN24:def 2
ex b1 being isometric Function of a1,a1 st b1 = a2;
end;

:: deftheorem Def2 defines isometric JORDAN24:def 2 :
for b1 being non empty MetrStruct
for b2 being Function of (TopSpaceMetr b1),(TopSpaceMetr b1) holds
( b2 is isometric iff ex b3 being isometric Function of b1,b1 st b3 = b2 );

registration
let c5 be non empty MetrStruct ;
cluster isometric Relation of the carrier of (TopSpaceMetr a1),the carrier of (TopSpaceMetr a1);
existence
ex b1 being Function of (TopSpaceMetr c5),(TopSpaceMetr c5) st b1 is isometric
proof end;
end;

registration
let c5 be non empty MetrSpace;
cluster isometric -> continuous Relation of the carrier of (TopSpaceMetr a1),the carrier of (TopSpaceMetr a1);
coherence
for b1 being Function of (TopSpaceMetr c5),(TopSpaceMetr c5) holds
( b1 is isometric implies b1 is continuous )
proof end;
end;

registration
let c5 be non empty MetrSpace;
cluster isometric -> being_homeomorphism Relation of the carrier of (TopSpaceMetr a1),the carrier of (TopSpaceMetr a1);
coherence
for b1 being Function of (TopSpaceMetr c5),(TopSpaceMetr c5) holds
( b1 is isometric implies b1 is being_homeomorphism )
proof end;
end;

definition
let c5 be Real;
func Rotate c1 -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def3: :: JORDAN24:def 3
for b1 being Point of (TOP-REAL 2) holds a2 . b1 = |[(Re (Rotate ((b1 `1 ) + ((b1 `2 ) * <i> )),a1)),(Im (Rotate ((b1 `1 ) + ((b1 `2 ) * <i> )),a1))]|;
existence
ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st
for b2 being Point of (TOP-REAL 2) holds b1 . b2 = |[(Re (Rotate ((b2 `1 ) + ((b2 `2 ) * <i> )),c5)),(Im (Rotate ((b2 `1 ) + ((b2 `2 ) * <i> )),c5))]|
proof end;
uniqueness
for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( ( for b3 being Point of (TOP-REAL 2) holds b1 . b3 = |[(Re (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5)),(Im (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5))]| ) & ( for b3 being Point of (TOP-REAL 2) holds b2 . b3 = |[(Re (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5)),(Im (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5))]| ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines Rotate JORDAN24:def 3 :
for b1 being Real
for b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( b2 = Rotate b1 iff for b3 being Point of (TOP-REAL 2) holds b2 . b3 = |[(Re (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),b1)),(Im (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),b1))]| );

theorem Th2: :: JORDAN24:2
for b1 being Real holds
( 0 <= b1 & b1 < 2 * PI implies for b2 being Function of (TopSpaceMetr (Euclid 2)),(TopSpaceMetr (Euclid 2)) holds
( b2 = Rotate b1 implies b2 is isometric ) )
proof end;

theorem Th3: :: JORDAN24:3
for b1, b2 being Point of (TOP-REAL 2)
for b3 being Subset of (TOP-REAL 2)
for b4, b5, b6 being real number holds
( b1,b2 realize-max-dist-in b3 implies (AffineMap b4,b5,b4,b6) . b1,(AffineMap b4,b5,b4,b6) . b2 realize-max-dist-in (AffineMap b4,b5,b4,b6) .: b3 )
proof end;

theorem Th4: :: JORDAN24:4
for b1, b2 being Point of (TOP-REAL 2)
for b3 being Subset of (TOP-REAL 2)
for b4 being Real holds
( 0 <= b4 & b4 < 2 * PI & b1,b2 realize-max-dist-in b3 implies (Rotate b4) . b1,(Rotate b4) . b2 realize-max-dist-in (Rotate b4) .: b3 )
proof end;

theorem Th5: :: JORDAN24:5
for b1 being complex number
for b2 being Real holds Rotate b1,(- b2) = Rotate b1,((2 * PI ) - b2)
proof end;

theorem Th6: :: JORDAN24:6
for b1 being Real holds Rotate (- b1) = Rotate ((2 * PI ) - b1)
proof end;

theorem Th7: :: JORDAN24:7
for b1 being Simple_closed_curve holds
ex b2 being Homeomorphism of TOP-REAL 2 st |[(- 1),0]|,|[1,0]| realize-max-dist-in b2 .: b1
proof end;

definition
let c5, c6 be TopStruct ;
let c7 be Function of c5,c6;
attr a3 is closed means :: JORDAN24:def 4
for b1 being Subset of a1 holds
( b1 is closed implies a3 .: b1 is closed );
end;

:: deftheorem Def4 defines closed JORDAN24:def 4 :
for b1, b2 being TopStruct
for b3 being Function of b1,b2 holds
( b3 is closed iff for b4 being Subset of b1 holds
( b4 is closed implies b3 .: b4 is closed ) );

theorem Th8: :: JORDAN24:8
for b1, b2 being non empty TopSpace
for b3 being continuous Function of b1,b2 holds
( b3 is one-to-one & b3 is onto implies ( b3 is_homeomorphism iff b3 is closed ) )
proof end;

theorem Th9: :: JORDAN24:9
for b1 being set
for b2 being Subset of b1 holds
( b2 ` = {} iff b2 = b1 )
proof end;

theorem Th10: :: JORDAN24:10
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism implies for b4 being Subset of b1 holds
( b4 is connected implies b3 .: b4 is connected ) )
proof end;

theorem Th11: :: JORDAN24:11
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism implies for b4 being Subset of b1 holds
( b4 is_a_component_of b1 implies b3 .: b4 is_a_component_of b2 ) )
proof end;

theorem Th12: :: JORDAN24:12
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2
for b4 being Subset of b1 holds
b3 | b4 is Function of (b1 | b4),(b2 | (b3 .: b4))
proof end;

theorem Th13: :: JORDAN24:13
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is continuous implies for b4 being Subset of b1
for b5 being Function of (b1 | b4),(b2 | (b3 .: b4)) holds
( b5 = b3 | b4 implies b5 is continuous ) )
proof end;

theorem Th14: :: JORDAN24:14
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism implies for b4 being Subset of b1
for b5 being Function of (b1 | b4),(b2 | (b3 .: b4)) holds
( b5 = b3 | b4 implies b5 is_homeomorphism ) )
proof end;

theorem Th15: :: JORDAN24:15
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism implies for b4, b5 being Subset of b1 holds
( b4 is_a_component_of b5 implies b3 .: b4 is_a_component_of b3 .: b5 ) )
proof end;

theorem Th16: :: JORDAN24:16
for b1 being Subset of (TOP-REAL 2)
for b2 being Homeomorphism of TOP-REAL 2 holds
( b1 is Jordan implies b2 .: b1 is Jordan )
proof end;