Article JORDAN24, MML version 4.99.1005

:: JORDAN24:prednot 1 => JORDAN24:pred 1
definition
  let a1 be Element of NAT;
  let a2 be Element of bool the carrier of TOP-REAL a1;
  let a3, a4 be Element of the carrier of TOP-REAL a1;
  pred A3,A4 realize-max-dist-in A2 means
    a3 in a2 &
     a4 in a2 &
     (for b1, b2 being Element of the carrier of TOP-REAL a1
           st b1 in a2 & b2 in a2
        holds dist(b1,b2) <= dist(a3,a4));
end;

:: JORDAN24:dfs 1
definiens
  let a1 be Element of NAT;
  let a2 be Element of bool the carrier of TOP-REAL a1;
  let a3, a4 be Element of the carrier of TOP-REAL a1;
To prove
     a3,a4 realize-max-dist-in a2
it is sufficient to prove
  thus a3 in a2 &
     a4 in a2 &
     (for b1, b2 being Element of the carrier of TOP-REAL a1
           st b1 in a2 & b2 in a2
        holds dist(b1,b2) <= dist(a3,a4));

:: JORDAN24:def 1
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1 holds
   b3,b4 realize-max-dist-in b2
iff
   b3 in b2 &
    b4 in b2 &
    (for b5, b6 being Element of the carrier of TOP-REAL b1
          st b5 in b2 & b6 in b2
       holds dist(b5,b6) <= dist(b3,b4));

:: JORDAN24:th 1
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   ex b2, b3 being Element of the carrier of TOP-REAL 2 st
      b2,b3 realize-max-dist-in b1;

:: JORDAN24:attrnot 1 => JORDAN24:attr 1
definition
  let a1 be non empty MetrStruct;
  let a2 be Function-like quasi_total Relation of the carrier of TopSpaceMetr a1,the carrier of TopSpaceMetr a1;
  attr a2 is isometric means
    ex b1 being Function-like quasi_total isometric Relation of the carrier of a1,the carrier of a1 st
       b1 = a2;
end;

:: JORDAN24:dfs 2
definiens
  let a1 be non empty MetrStruct;
  let a2 be Function-like quasi_total Relation of the carrier of TopSpaceMetr a1,the carrier of TopSpaceMetr a1;
To prove
     a2 is isometric
it is sufficient to prove
  thus ex b1 being Function-like quasi_total isometric Relation of the carrier of a1,the carrier of a1 st
       b1 = a2;

:: JORDAN24:def 2
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of the carrier of TopSpaceMetr b1,the carrier of TopSpaceMetr b1 holds
      b2 is isometric(b1)
   iff
      ex b3 being Function-like quasi_total isometric Relation of the carrier of b1,the carrier of b1 st
         b3 = b2;

:: JORDAN24:exreg 1
registration
  let a1 be non empty MetrStruct;
  cluster Relation-like Function-like non empty quasi_total total isometric Relation of the carrier of TopSpaceMetr a1,the carrier of TopSpaceMetr a1;
end;

:: JORDAN24:condreg 1
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster Function-like quasi_total isometric -> continuous (Relation of the carrier of TopSpaceMetr a1,the carrier of TopSpaceMetr a1);
end;

:: JORDAN24:condreg 2
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster Function-like quasi_total isometric -> being_homeomorphism (Relation of the carrier of TopSpaceMetr a1,the carrier of TopSpaceMetr a1);
end;

:: JORDAN24:funcnot 1 => JORDAN24:func 1
definition
  let a1 be Element of REAL;
  func Rotate A1 -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 means
    for b1 being Element of the carrier of TOP-REAL 2 holds
       it . b1 = |[Re Rotate(b1 `1 + (b1 `2 * <i>),a1),Im Rotate(b1 `1 + (b1 `2 * <i>),a1)]|;
end;

:: JORDAN24:def 3
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 holds
      b2 = Rotate b1
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
         b2 . b3 = |[Re Rotate(b3 `1 + (b3 `2 * <i>),b1),Im Rotate(b3 `1 + (b3 `2 * <i>),b1)]|;

:: JORDAN24:th 2
theorem
for b1 being Element of REAL
   st 0 <= b1 & b1 < 2 * PI
for b2 being Function-like quasi_total Relation of the carrier of TopSpaceMetr Euclid 2,the carrier of TopSpaceMetr Euclid 2
      st b2 = Rotate b1
   holds b2 is isometric(Euclid 2);

:: JORDAN24:th 3
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of TOP-REAL 2
for b4, b5, b6 being real set
      st b1,b2 realize-max-dist-in b3
   holds (AffineMap(b4,b5,b4,b6)) . b1,(AffineMap(b4,b5,b4,b6)) . b2 realize-max-dist-in (AffineMap(b4,b5,b4,b6)) .: b3;

:: JORDAN24:th 4
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Element of REAL
      st 0 <= b4 & b4 < 2 * PI & b1,b2 realize-max-dist-in b3
   holds (Rotate b4) . b1,(Rotate b4) . b2 realize-max-dist-in (Rotate b4) .: b3;

:: JORDAN24:th 5
theorem
for b1 being complex set
for b2 being Element of REAL holds
   Rotate(b1,- b2) = Rotate(b1,(2 * PI) - b2);

:: JORDAN24:th 6
theorem
for b1 being Element of REAL holds
   Rotate - b1 = Rotate ((2 * PI) - b1);

:: JORDAN24:th 7
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   ex b2 being Homeomorphism of TOP-REAL 2 st
      |[- 1,0]|,|[1,0]| realize-max-dist-in b2 .: b1;

:: JORDAN24:attrnot 2 => JORDAN24:attr 2
definition
  let a1, a2 be TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is closed means
    for b1 being Element of bool the carrier of a1
          st b1 is closed(a1)
       holds a3 .: b1 is closed(a2);
end;

:: JORDAN24:dfs 4
definiens
  let a1, a2 be TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is closed
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 is closed(a1)
       holds a3 .: b1 is closed(a2);

:: JORDAN24:def 4
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is closed(b1, b2)
   iff
      for b4 being Element of bool the carrier of b1
            st b4 is closed(b1)
         holds b3 .: b4 is closed(b2);

:: JORDAN24:th 8
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
      st b3 is one-to-one & b3 is onto(the carrier of b1, the carrier of b2)
   holds    b3 is being_homeomorphism(b1, b2)
   iff
      b3 is closed(b1, b2);

:: JORDAN24:th 9
theorem
for b1 being set
for b2 being Element of bool b1 holds
      b2 ` = {}
   iff
      b2 = b1;

:: JORDAN24:th 10
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is being_homeomorphism(b1, b2)
for b4 being Element of bool the carrier of b1
      st b4 is connected(b1)
   holds b3 .: b4 is connected(b2);

:: JORDAN24:th 11
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is being_homeomorphism(b1, b2)
for b4 being Element of bool the carrier of b1
      st b4 is_a_component_of b1
   holds b3 .: b4 is_a_component_of b2;

:: JORDAN24:th 12
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1 holds
   b3 | b4 is Function-like quasi_total Relation of the carrier of b1 | b4,the carrier of b2 | (b3 .: b4);

:: JORDAN24:th 13
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is continuous(b1, b2)
for b4 being Element of bool the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1 | b4,the carrier of b2 | (b3 .: b4)
      st b5 = b3 | b4
   holds b5 is continuous(b1 | b4, b2 | (b3 .: b4));

:: JORDAN24:th 14
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is being_homeomorphism(b1, b2)
for b4 being Element of bool the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1 | b4,the carrier of b2 | (b3 .: b4)
      st b5 = b3 | b4
   holds b5 is being_homeomorphism(b1 | b4, b2 | (b3 .: b4));

:: JORDAN24:th 15
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is being_homeomorphism(b1, b2)
for b4, b5 being Element of bool the carrier of b1
      st b4 is_a_component_of b5
   holds b3 .: b4 is_a_component_of b3 .: b5;

:: JORDAN24:th 16
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Homeomorphism of TOP-REAL 2
      st b1 is Jordan
   holds b2 .: b1 is Jordan;