Article TOPREALB, MML version 4.99.1005

:: TOPREALB:funcreg 1
registration
  cluster ].0,1.[ -> non empty;
end;

:: TOPREALB:funcreg 2
registration
  cluster [.- 1,1.] -> non empty;
end;

:: TOPREALB:funcreg 3
registration
  cluster ].1 / 2,3 / 2.[ -> non empty;
end;

:: TOPREALB:funcreg 4
registration
  cluster sin -> Function-like quasi_total continuous;
end;

:: TOPREALB:funcreg 5
registration
  cluster cos -> Function-like quasi_total continuous;
end;

:: TOPREALB:funcreg 6
registration
  cluster arcsin -> Function-like continuous;
end;

:: TOPREALB:funcreg 7
registration
  cluster arccos -> Function-like continuous;
end;

:: TOPREALB:th 1
theorem
for b1, b2, b3 being real set holds
sin ((b1 * b2) + b3) = (sin * AffineMap(b1,b3)) . b2;

:: TOPREALB:th 2
theorem
for b1, b2, b3 being real set holds
cos ((b1 * b2) + b3) = (cos * AffineMap(b1,b3)) . b2;

:: TOPREALB:funcreg 8
registration
  let a1 be non empty real set;
  let a2 be real set;
  cluster AffineMap(a1,a2) -> Function-like one-to-one quasi_total onto;
end;

:: TOPREALB:funcnot 1 => TOPREALB:func 1
definition
  let a1, a2 be real set;
  func IntIntervals(A1,A2) -> set equals
    {].a1 + b1,a2 + b1.[ where b1 is Element of INT: TRUE};
end;

:: TOPREALB:def 1
theorem
for b1, b2 being real set holds
IntIntervals(b1,b2) = {].b1 + b3,b2 + b3.[ where b3 is Element of INT: TRUE};

:: TOPREALB:th 3
theorem
for b1, b2 being real set
for b3 being set holds
      b3 in IntIntervals(b1,b2)
   iff
      ex b4 being Element of INT st
         b3 = ].b1 + b4,b2 + b4.[;

:: TOPREALB:funcreg 9
registration
  let a1, a2 be real set;
  cluster IntIntervals(a1,a2) -> non empty;
end;

:: TOPREALB:th 4
theorem
for b1, b2 being real set
      st b1 - b2 <= 1
   holds IntIntervals(b2,b1) is mutually-disjoint;

:: TOPREALB:funcnot 2 => TOPREALB:func 2
definition
  let a1, a2 be real set;
  redefine func IntIntervals(a1,a2) -> Element of bool bool the carrier of R^1;
end;

:: TOPREALB:funcnot 3 => TOPREALB:func 3
definition
  let a1, a2 be real set;
  redefine func IntIntervals(a1,a2) -> open Element of bool bool the carrier of R^1;
end;

:: TOPREALB:funcnot 4 => TOPREALB:func 4
definition
  let a1 be real set;
  func R^1 A1 -> Element of the carrier of R^1 equals
    a1;
end;

:: TOPREALB:def 2
theorem
for b1 being real set holds
   R^1 b1 = b1;

:: TOPREALB:funcnot 5 => TOPREALB:func 5
definition
  let a1 be Element of bool REAL;
  func R^1 A1 -> Element of bool the carrier of R^1 equals
    a1;
end;

:: TOPREALB:def 3
theorem
for b1 being Element of bool REAL holds
   R^1 b1 = b1;

:: TOPREALB:funcreg 10
registration
  let a1 be non empty Element of bool REAL;
  cluster R^1 a1 -> non empty;
end;

:: TOPREALB:funcreg 11
registration
  let a1 be open Element of bool REAL;
  cluster R^1 a1 -> open;
end;

:: TOPREALB:funcreg 12
registration
  let a1 be closed Element of bool REAL;
  cluster R^1 a1 -> closed;
end;

:: TOPREALB:funcreg 13
registration
  let a1 be open Element of bool REAL;
  cluster R^1 | R^1 a1 -> strict open;
end;

:: TOPREALB:funcreg 14
registration
  let a1 be closed Element of bool REAL;
  cluster R^1 | R^1 a1 -> strict closed;
end;

:: TOPREALB:funcnot 6 => TOPREALB:func 6
definition
  let a1 be Function-like Relation of REAL,REAL;
  func R^1 A1 -> Function-like quasi_total Relation of the carrier of R^1 | R^1 dom a1,the carrier of R^1 | R^1 rng a1 equals
    a1;
end;

:: TOPREALB:def 4
theorem
for b1 being Function-like Relation of REAL,REAL holds
   R^1 b1 = b1;

:: TOPREALB:funcreg 15
registration
  let a1 be Function-like Relation of REAL,REAL;
  cluster R^1 a1 -> Function-like quasi_total onto;
end;

:: TOPREALB:funcreg 16
registration
  let a1 be Function-like one-to-one Relation of REAL,REAL;
  cluster R^1 a1 -> Function-like one-to-one quasi_total;
end;

:: TOPREALB:th 5
theorem
R^1 | R^1 [#] REAL = R^1;

:: TOPREALB:th 6
theorem
for b1 being Function-like Relation of REAL,REAL
      st dom b1 = REAL
   holds R^1 | R^1 dom b1 = R^1;

:: TOPREALB:th 7
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL holds
   b1 is Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1 | R^1 rng b1;

:: TOPREALB:th 8
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL holds
   b1 is Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1;

:: TOPREALB:funcreg 17
registration
  let a1 be Function-like continuous Relation of REAL,REAL;
  cluster R^1 a1 -> Function-like quasi_total continuous;
end;

:: TOPREALB:funcreg 18
registration
  let a1 be non empty real set;
  let a2 be real set;
  cluster R^1 AffineMap(a1,a2) -> Function-like quasi_total open;
end;

:: TOPREALB:attrnot 1 => TOPREALB:attr 1
definition
  let a1 be SubSpace of TOP-REAL 2;
  attr a1 is being_simple_closed_curve means
    the carrier of a1 is being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
end;

:: TOPREALB:dfs 5
definiens
  let a1 be SubSpace of TOP-REAL 2;
To prove
     a1 is being_simple_closed_curve
it is sufficient to prove
  thus the carrier of a1 is being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;

:: TOPREALB:def 5
theorem
for b1 being SubSpace of TOP-REAL 2 holds
      b1 is being_simple_closed_curve
   iff
      the carrier of b1 is being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;

:: TOPREALB:condreg 1
registration
  cluster being_simple_closed_curve -> non empty compact arcwise_connected (SubSpace of TOP-REAL 2);
end;

:: TOPREALB:funcreg 19
registration
  let a1 be real positive set;
  let a2 be Element of the carrier of TOP-REAL 2;
  cluster Sphere(a2,a1) -> being_simple_closed_curve;
end;

:: TOPREALB:funcnot 7 => TOPREALB:func 7
definition
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  func Tcircle(A2,A3) -> SubSpace of TOP-REAL a1 equals
    (TOP-REAL a1) | Sphere(a2,a3);
end;

:: TOPREALB:def 6
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being real set holds
   Tcircle(b2,b3) = (TOP-REAL b1) | Sphere(b2,b3);

:: TOPREALB:funcreg 20
registration
  let a1 be non empty Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real non negative set;
  cluster Tcircle(a2,a3) -> non empty strict;
end;

:: TOPREALB:th 9
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1 holds
   the carrier of Tcircle(b3,b2) = Sphere(b3,b2);

:: TOPREALB:funcreg 21
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be empty real set;
  cluster Tcircle(a2,a3) -> trivial;
end;

:: TOPREALB:th 10
theorem
for b1 being real set holds
   Tcircle(0.REAL 2,b1) is SubSpace of Trectangle(- b1,b1,- b1,b1);

:: TOPREALB:funcreg 22
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  let a2 be real positive set;
  cluster Tcircle(a1,a2) -> being_simple_closed_curve;
end;

:: TOPREALB:exreg 1
registration
  cluster strict TopSpace-like constituted-Functions constituted-FinSeqs being_simple_closed_curve SubSpace of TOP-REAL 2;
end;

:: TOPREALB:th 11
theorem
for b1, b2 being being_simple_closed_curve SubSpace of TOP-REAL 2 holds
b1,b2 are_homeomorphic;

:: TOPREALB:funcnot 8 => TOPREALB:func 8
definition
  let a1 be Element of NAT;
  func Tunit_circle A1 -> SubSpace of TOP-REAL a1 equals
    Tcircle(0.REAL a1,1);
end;

:: TOPREALB:def 7
theorem
for b1 being Element of NAT holds
   Tunit_circle b1 = Tcircle(0.REAL b1,1);

:: TOPREALB:funcreg 23
registration
  let a1 be non empty Element of NAT;
  cluster Tunit_circle a1 -> non empty;
end;

:: TOPREALB:th 12
theorem
for b1 being non empty Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
      st b2 is Element of the carrier of Tunit_circle b1
   holds |.b2.| = 1;

:: TOPREALB:th 13
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Tunit_circle 2
   holds - 1 <= b1 `1 & b1 `1 <= 1 & - 1 <= b1 `2 & b1 `2 <= 1;

:: TOPREALB:th 14
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Tunit_circle 2 & b1 `1 = 1
   holds b1 `2 = 0;

:: TOPREALB:th 15
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Tunit_circle 2 & b1 `1 = - 1
   holds b1 `2 = 0;

:: TOPREALB:th 16
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Tunit_circle 2 & b1 `2 = 1
   holds b1 `1 = 0;

:: TOPREALB:th 17
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Tunit_circle 2 & b1 `2 = - 1
   holds b1 `1 = 0;

:: TOPREALB:th 18
theorem
Tunit_circle 2 is SubSpace of Trectangle(- 1,1,- 1,1);

:: TOPREALB:th 19
theorem
for b1 being non empty Element of NAT
for b2 being real positive set
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Function-like quasi_total Relation of the carrier of Tunit_circle b1,the carrier of Tcircle(b3,b2)
      st for b5 being Element of the carrier of Tunit_circle b1
        for b6 being Element of the carrier of TOP-REAL b1
              st b5 = b6
           holds b4 . b5 = (b2 * b6) + b3
   holds b4 is being_homeomorphism(Tunit_circle b1, Tcircle(b3,b2));

:: TOPREALB:funcreg 24
registration
  cluster Tunit_circle 2 -> being_simple_closed_curve;
end;

:: TOPREALB:th 20
theorem
for b1 being non empty Element of NAT
for b2, b3 being real positive set
for b4, b5 being Element of the carrier of TOP-REAL b1 holds
Tcircle(b4,b2),Tcircle(b5,b3) are_homeomorphic;

:: TOPREALB:funcreg 25
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  let a2 be real non negative set;
  cluster Tcircle(a1,a2) -> arcwise_connected;
end;

:: TOPREALB:funcnot 9 => TOPREALB:func 9
definition
  func c[10] -> Element of the carrier of Tunit_circle 2 equals
    |[1,0]|;
end;

:: TOPREALB:def 8
theorem
c[10] = |[1,0]|;

:: TOPREALB:funcnot 10 => TOPREALB:func 10
definition
  func c[-10] -> Element of the carrier of Tunit_circle 2 equals
    |[- 1,0]|;
end;

:: TOPREALB:def 9
theorem
c[-10] = |[- 1,0]|;

:: TOPREALB:funcnot 11 => TOPREALB:func 11
definition
  let a1 be Element of the carrier of Tunit_circle 2;
  func Topen_unit_circle A1 -> strict SubSpace of Tunit_circle 2 means
    the carrier of it = (the carrier of Tunit_circle 2) \ {a1};
end;

:: TOPREALB:def 10
theorem
for b1 being Element of the carrier of Tunit_circle 2
for b2 being strict SubSpace of Tunit_circle 2 holds
      b2 = Topen_unit_circle b1
   iff
      the carrier of b2 = (the carrier of Tunit_circle 2) \ {b1};

:: TOPREALB:funcreg 26
registration
  let a1 be Element of the carrier of Tunit_circle 2;
  cluster Topen_unit_circle a1 -> non empty strict;
end;

:: TOPREALB:th 22
theorem
for b1 being Element of the carrier of Tunit_circle 2 holds
   b1 is not Element of the carrier of Topen_unit_circle b1;

:: TOPREALB:th 23
theorem
for b1 being Element of the carrier of Tunit_circle 2 holds
   Topen_unit_circle b1 = (Tunit_circle 2) | (([#] Tunit_circle 2) \ {b1});

:: TOPREALB:th 24
theorem
for b1, b2 being Element of the carrier of Tunit_circle 2
      st b1 <> b2
   holds b2 is Element of the carrier of Topen_unit_circle b1;

:: TOPREALB:th 25
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Topen_unit_circle c[10] & b1 `2 = 0
   holds b1 = c[-10];

:: TOPREALB:th 26
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Topen_unit_circle c[-10] & b1 `2 = 0
   holds b1 = c[10];

:: TOPREALB:th 27
theorem
for b1 being Element of the carrier of Tunit_circle 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 is Element of the carrier of Topen_unit_circle b1
   holds - 1 <= b2 `1 & b2 `1 <= 1 & - 1 <= b2 `2 & b2 `2 <= 1;

:: TOPREALB:th 28
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Topen_unit_circle c[10]
   holds - 1 <= b1 `1 & b1 `1 < 1;

:: TOPREALB:th 29
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 is Element of the carrier of Topen_unit_circle c[-10]
   holds - 1 < b1 `1 & b1 `1 <= 1;

:: TOPREALB:funcreg 27
registration
  let a1 be Element of the carrier of Tunit_circle 2;
  cluster Topen_unit_circle a1 -> strict open;
end;

:: TOPREALB:th 30
theorem
for b1 being Element of the carrier of Tunit_circle 2 holds
   Topen_unit_circle b1,I(01) are_homeomorphic;

:: TOPREALB:th 31
theorem
for b1, b2 being Element of the carrier of Tunit_circle 2 holds
Topen_unit_circle b1,Topen_unit_circle b2 are_homeomorphic;

:: TOPREALB:funcnot 12 => TOPREALB:func 12
definition
  func CircleMap -> Function-like quasi_total Relation of the carrier of R^1,the carrier of Tunit_circle 2 means
    for b1 being real set holds
       it . b1 = |[cos ((2 * PI) * b1),sin ((2 * PI) * b1)]|;
end;

:: TOPREALB:def 11
theorem
for b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of Tunit_circle 2 holds
      b1 = CircleMap
   iff
      for b2 being real set holds
         b1 . b2 = |[cos ((2 * PI) * b2),sin ((2 * PI) * b2)]|;

:: TOPREALB:th 32
theorem
for b1 being integer set
for b2 being real set holds
   CircleMap . b2 = CircleMap . (b2 + b1);

:: TOPREALB:th 33
theorem
for b1 being integer set holds
   CircleMap . b1 = c[10];

:: TOPREALB:th 34
theorem
CircleMap " {c[10]} = INT;

:: TOPREALB:th 35
theorem
for b1 being real set
      st frac b1 = 1 / 2
   holds CircleMap . b1 = |[- 1,0]|;

:: TOPREALB:th 36
theorem
for b1 being real set
      st frac b1 = 1 / 4
   holds CircleMap . b1 = |[0,1]|;

:: TOPREALB:th 37
theorem
for b1 being real set
      st frac b1 = 3 / 4
   holds CircleMap . b1 = |[0,- 1]|;

:: TOPREALB:th 38
theorem
for b1 being real set
for b2, b3 being integer set holds
CircleMap . b1 = |[(cos * AffineMap(2 * PI,(2 * PI) * b2)) . b1,(sin * AffineMap(2 * PI,(2 * PI) * b3)) . b1]|;

:: TOPREALB:funcreg 28
registration
  cluster CircleMap -> Function-like quasi_total continuous;
end;

:: TOPREALB:th 39
theorem
for b1 being Element of bool the carrier of R^1
for b2 being Function-like quasi_total Relation of the carrier of R^1 | b1,the carrier of Tunit_circle 2
      st [.0,1.[ c= b1 & b2 = CircleMap | b1
   holds b2 is onto(the carrier of R^1 | b1, the carrier of Tunit_circle 2);

:: TOPREALB:funcreg 29
registration
  cluster CircleMap -> Function-like quasi_total onto;
end;

:: TOPREALB:funcreg 30
registration
  let a1 be real set;
  cluster CircleMap | [.a1,a1 + 1.[ -> Relation-like one-to-one;
end;

:: TOPREALB:funcreg 31
registration
  let a1 be real set;
  cluster CircleMap | ].a1,a1 + 1.[ -> Relation-like one-to-one;
end;

:: TOPREALB:th 40
theorem
for b1, b2 being real set
   st b1 - b2 <= 1
for b3 being set
      st b3 in IntIntervals(b2,b1)
   holds CircleMap | b3 is one-to-one;

:: TOPREALB:th 41
theorem
for b1, b2 being real set
for b3 being set
      st b3 in IntIntervals(b1,b2)
   holds CircleMap .: b3 = CircleMap .: union IntIntervals(b1,b2);

:: TOPREALB:funcnot 13 => TOPREALB:func 13
definition
  let a1 be Element of the carrier of R^1;
  func CircleMap A1 -> Function-like quasi_total Relation of the carrier of R^1 | R^1 ].a1,a1 + 1.[,the carrier of Topen_unit_circle (CircleMap . a1) equals
    CircleMap | ].a1,a1 + 1.[;
end;

:: TOPREALB:def 12
theorem
for b1 being Element of the carrier of R^1 holds
   CircleMap b1 = CircleMap | ].b1,b1 + 1.[;

:: TOPREALB:th 42
theorem
for b1 being integer set
for b2 being real set holds
   CircleMap R^1 (b2 + b1) = (CircleMap R^1 b2) * ((AffineMap(1,- b1)) | ].b2 + b1,(b2 + b1) + 1.[);

:: TOPREALB:funcreg 32
registration
  let a1 be Element of the carrier of R^1;
  cluster CircleMap a1 -> Function-like one-to-one quasi_total onto continuous;
end;

:: TOPREALB:funcnot 14 => TOPREALB:func 14
definition
  func Circle2IntervalR -> Function-like quasi_total Relation of the carrier of Topen_unit_circle c[10],the carrier of R^1 | R^1 ].0,1.[ means
    for b1 being Element of the carrier of Topen_unit_circle c[10] holds
       ex b2, b3 being real set st
          b1 = |[b2,b3]| &
           (0 <= b3 implies it . b1 = (arccos b2) / (2 * PI)) &
           (b3 <= 0 implies it . b1 = 1 - ((arccos b2) / (2 * PI)));
end;

:: TOPREALB:def 13
theorem
for b1 being Function-like quasi_total Relation of the carrier of Topen_unit_circle c[10],the carrier of R^1 | R^1 ].0,1.[ holds
      b1 = Circle2IntervalR
   iff
      for b2 being Element of the carrier of Topen_unit_circle c[10] holds
         ex b3, b4 being real set st
            b2 = |[b3,b4]| &
             (0 <= b4 implies b1 . b2 = (arccos b3) / (2 * PI)) &
             (b4 <= 0 implies b1 . b2 = 1 - ((arccos b3) / (2 * PI)));

:: TOPREALB:funcnot 15 => TOPREALB:func 15
definition
  func Circle2IntervalL -> Function-like quasi_total Relation of the carrier of Topen_unit_circle c[-10],the carrier of R^1 | R^1 ].1 / 2,3 / 2.[ means
    for b1 being Element of the carrier of Topen_unit_circle c[-10] holds
       ex b2, b3 being real set st
          b1 = |[b2,b3]| &
           (0 <= b3 implies it . b1 = 1 + ((arccos b2) / (2 * PI))) &
           (b3 <= 0 implies it . b1 = 1 - ((arccos b2) / (2 * PI)));
end;

:: TOPREALB:def 14
theorem
for b1 being Function-like quasi_total Relation of the carrier of Topen_unit_circle c[-10],the carrier of R^1 | R^1 ].1 / 2,3 / 2.[ holds
      b1 = Circle2IntervalL
   iff
      for b2 being Element of the carrier of Topen_unit_circle c[-10] holds
         ex b3, b4 being real set st
            b2 = |[b3,b4]| &
             (0 <= b4 implies b1 . b2 = 1 + ((arccos b3) / (2 * PI))) &
             (b4 <= 0 implies b1 . b2 = 1 - ((arccos b3) / (2 * PI)));

:: TOPREALB:th 43
theorem
(CircleMap R^1 0) /" = Circle2IntervalR;

:: TOPREALB:th 44
theorem
(CircleMap R^1 (1 / 2)) /" = Circle2IntervalL;

:: TOPREALB:funcreg 33
registration
  cluster Circle2IntervalR -> Function-like one-to-one quasi_total onto continuous;
end;

:: TOPREALB:funcreg 34
registration
  cluster Circle2IntervalL -> Function-like one-to-one quasi_total onto continuous;
end;

:: TOPREALB:funcreg 35
registration
  let a1 be integer set;
  cluster CircleMap R^1 a1 -> Function-like quasi_total open;
end;

:: TOPREALB:funcreg 36
registration
  let a1 be integer set;
  cluster CircleMap R^1 ((1 / 2) + a1) -> Function-like quasi_total open;
end;

:: TOPREALB:funcreg 37
registration
  cluster Circle2IntervalR -> Function-like quasi_total open;
end;

:: TOPREALB:funcreg 38
registration
  cluster Circle2IntervalL -> Function-like quasi_total open;
end;

:: TOPREALB:th 46
theorem
CircleMap R^1 (1 / 2) is being_homeomorphism(R^1 | R^1 ].R^1 (1 / 2),(R^1 (1 / 2)) + 1.[, Topen_unit_circle (CircleMap . R^1 (1 / 2)));

:: TOPREALB:th 49
theorem
ex b1 being Element of bool bool the carrier of Tunit_circle 2 st
   b1 = {CircleMap .: ].0,1.[,CircleMap .: ].1 / 2,3 / 2.[} &
    b1 is_a_cover_of Tunit_circle 2 &
    b1 is open(Tunit_circle 2) &
    (for b2 being Element of bool the carrier of Tunit_circle 2 holds
       (b2 = CircleMap .: ].0,1.[ implies union IntIntervals(0,1) = CircleMap " b2 &
         (for b3 being Element of bool the carrier of R^1
            st b3 in IntIntervals(0,1)
         for b4 being Function-like quasi_total Relation of the carrier of R^1 | b3,the carrier of (Tunit_circle 2) | b2
               st b4 = CircleMap | b3
            holds b4 is being_homeomorphism(R^1 | b3, (Tunit_circle 2) | b2))) &
        (b2 = CircleMap .: ].1 / 2,3 / 2.[ implies union IntIntervals(1 / 2,3 / 2) = CircleMap " b2 &
         (for b3 being Element of bool the carrier of R^1
            st b3 in IntIntervals(1 / 2,3 / 2)
         for b4 being Function-like quasi_total Relation of the carrier of R^1 | b3,the carrier of (Tunit_circle 2) | b2
               st b4 = CircleMap | b3
            holds b4 is being_homeomorphism(R^1 | b3, (Tunit_circle 2) | b2))));