Article BORSUK_6, MML version 4.99.1005

:: BORSUK_6:sch 1
scheme BORSUK_6:sch 1
{F1 -> non empty set,
  F2 -> set,
  F3 -> set,
  F4 -> set}:
ex b1 being Relation-like Function-like set st
   proj1 b1 = F1() &
    (for b2 being Element of F1() holds
       (P1[b2] implies b1 . b2 = F2(b2)) & (P2[b2] implies b1 . b2 = F3(b2)) & (P3[b2] implies b1 . b2 = F4(b2)))
provided
   for b1 being Element of F1() holds
      (P1[b1] implies not (P2[b1])) & (P1[b1] implies not (P3[b1])) & (P2[b1] implies not (P3[b1]))
and
   for b1 being Element of F1()
         st not (P1[b1]) & not (P2[b1])
      holds P3[b1];


:: BORSUK_6:th 1
theorem
the carrier of [:I[01],I[01]:] = [:[.0,1.],[.0,1.]:];

:: BORSUK_6:th 5
theorem
for b1, b2, b3 being real set
      st b1 <= b3 & b3 <= b2
   holds (b3 - b1) / (b2 - b1) in the carrier of Closed-Interval-TSpace(0,1);

:: BORSUK_6:th 6
theorem
for b1 being Element of the carrier of I[01]
      st b1 <= 1 / 2
   holds 2 * b1 is Element of the carrier of I[01];

:: BORSUK_6:th 7
theorem
for b1 being Element of the carrier of I[01]
      st 1 / 2 <= b1
   holds (2 * b1) - 1 is Element of the carrier of I[01];

:: BORSUK_6:th 8
theorem
for b1, b2 being Element of the carrier of I[01] holds
b1 * b2 is Element of the carrier of I[01];

:: BORSUK_6:th 9
theorem
for b1 being Element of the carrier of I[01] holds
   (1 / 2) * b1 is Element of the carrier of I[01];

:: BORSUK_6:th 10
theorem
for b1 being Element of the carrier of I[01]
      st 1 / 2 <= b1
   holds b1 - (1 / 4) is Element of the carrier of I[01];

:: BORSUK_6:th 12
theorem
id I[01] is Path of 0[01],1[01];

:: BORSUK_6:th 13
theorem
for b1, b2, b3, b4 being Element of the carrier of I[01]
      st b1 <= b2 & b3 <= b4
   holds [:[.b1,b2.],[.b3,b4.]:] is non empty compact Element of bool the carrier of [:I[01],I[01]:];

:: BORSUK_6:th 14
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= (2 * (b3 `1)) - 1} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= b3 `1}
   holds (AffineMap(1,0,1 / 2,1 / 2)) .: b1 = b2;

:: BORSUK_6:th 15
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: (2 * (b3 `1)) - 1 <= b3 `2} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 <= b3 `2}
   holds (AffineMap(1,0,1 / 2,1 / 2)) .: b1 = b2;

:: BORSUK_6:th 16
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: 1 - (2 * (b3 `1)) <= b3 `2} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: - (b3 `1) <= b3 `2}
   holds (AffineMap(1,0,1 / 2,- (1 / 2))) .: b1 = b2;

:: BORSUK_6:th 17
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= 1 - (2 * (b3 `1))} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= - (b3 `1)}
   holds (AffineMap(1,0,1 / 2,- (1 / 2))) .: b1 = b2;

:: BORSUK_6:th 18
theorem
for b1 being non empty 1-sorted holds
      b1 is real-membered
   iff
      for b2 being Element of the carrier of b1 holds
         b2 is real;

:: BORSUK_6:exreg 1
registration
  cluster non empty real-membered 1-sorted;
end;

:: BORSUK_6:exreg 2
registration
  cluster non empty TopSpace-like real-membered TopStruct;
end;

:: BORSUK_6:condreg 1
registration
  let a1 be real-membered 1-sorted;
  cluster -> real (Element of the carrier of a1);
end;

:: BORSUK_6:condreg 2
registration
  let a1 be real-membered TopStruct;
  cluster -> real-membered (SubSpace of a1);
end;

:: BORSUK_6:funcreg 1
registration
  let a1, a2 be non empty TopSpace-like real-membered TopStruct;
  let a3 be Element of the carrier of [:a1,a2:];
  cluster a3 `1 -> real;
end;

:: BORSUK_6:funcreg 2
registration
  let a1, a2 be non empty TopSpace-like real-membered TopStruct;
  let a3 be Element of the carrier of [:a1,a2:];
  cluster a3 `2 -> real;
end;

:: BORSUK_6:funcreg 3
registration
  let a1 be non empty SubSpace of [:I[01],I[01]:];
  let a2 be Element of the carrier of a1;
  cluster a2 `1 -> real;
end;

:: BORSUK_6:funcreg 4
registration
  let a1 be non empty SubSpace of [:I[01],I[01]:];
  let a2 be Element of the carrier of a1;
  cluster a2 `2 -> real;
end;

:: BORSUK_6:funcreg 5
registration
  cluster R^1 -> strict TopSpace-like real-membered;
end;

:: BORSUK_6:th 19
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `2 <= (2 * (b1 `1)) - 1} is closed Element of bool the carrier of TOP-REAL 2;

:: BORSUK_6:th 20
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: (2 * (b1 `1)) - 1 <= b1 `2} is closed Element of bool the carrier of TOP-REAL 2;

:: BORSUK_6:th 21
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `2 <= 1 - (2 * (b1 `1))} is closed Element of bool the carrier of TOP-REAL 2;

:: BORSUK_6:th 22
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: 1 - (2 * (b1 `1)) <= b1 `2} is closed Element of bool the carrier of TOP-REAL 2;

:: BORSUK_6:th 23
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: 1 - (2 * (b1 `1)) <= b1 `2 &
 (2 * (b1 `1)) - 1 <= b1 `2} is closed Element of bool the carrier of TOP-REAL 2;

:: BORSUK_6:th 24
theorem
ex b1 being Function-like quasi_total Relation of the carrier of [:R^1,R^1:],the carrier of TOP-REAL 2 st
   for b2, b3 being Element of REAL holds
   b1 . [b2,b3] = <*b2,b3*>;

:: BORSUK_6:th 25
theorem
{b1 where b1 is Element of the carrier of [:R^1,R^1:]: b1 `2 <= 1 - (2 * (b1 `1))} is closed Element of bool the carrier of [:R^1,R^1:];

:: BORSUK_6:th 26
theorem
{b1 where b1 is Element of the carrier of [:R^1,R^1:]: b1 `2 <= (2 * (b1 `1)) - 1} is closed Element of bool the carrier of [:R^1,R^1:];

:: BORSUK_6:th 27
theorem
{b1 where b1 is Element of the carrier of [:R^1,R^1:]: 1 - (2 * (b1 `1)) <= b1 `2 &
 (2 * (b1 `1)) - 1 <= b1 `2} is closed Element of bool the carrier of [:R^1,R^1:];

:: BORSUK_6:th 28
theorem
{b1 where b1 is Element of the carrier of [:I[01],I[01]:]: b1 `2 <= 1 - (2 * (b1 `1))} is non empty closed Element of bool the carrier of [:I[01],I[01]:];

:: BORSUK_6:th 29
theorem
{b1 where b1 is Element of the carrier of [:I[01],I[01]:]: 1 - (2 * (b1 `1)) <= b1 `2 &
 (2 * (b1 `1)) - 1 <= b1 `2} is non empty closed Element of bool the carrier of [:I[01],I[01]:];

:: BORSUK_6:th 30
theorem
{b1 where b1 is Element of the carrier of [:I[01],I[01]:]: b1 `2 <= (2 * (b1 `1)) - 1} is non empty closed Element of bool the carrier of [:I[01],I[01]:];

:: BORSUK_6:th 31
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of [:b1,b2:] holds
   b3 `1 is Element of the carrier of b1 & b3 `2 is Element of the carrier of b2;

:: BORSUK_6:th 32
theorem
for b1, b2 being Element of bool the carrier of [:I[01],I[01]:]
      st b1 = [:[.0,1 / 2.],[.0,1.]:] &
         b2 = [:[.1 / 2,1.],[.0,1.]:]
   holds ([#] ([:I[01],I[01]:] | b1)) \/ [#] ([:I[01],I[01]:] | b2) = [#] [:I[01],I[01]:];

:: BORSUK_6:th 33
theorem
for b1, b2 being Element of bool the carrier of [:I[01],I[01]:]
      st b1 = [:[.0,1 / 2.],[.0,1.]:] &
         b2 = [:[.1 / 2,1.],[.0,1.]:]
   holds ([#] ([:I[01],I[01]:] | b1)) /\ [#] ([:I[01],I[01]:] | b2) = [:{1 / 2},[.0,1.]:];

:: BORSUK_6:funcreg 6
registration
  let a1 be TopStruct;
  cluster {} a1 -> compact;
end;

:: BORSUK_6:exreg 3
registration
  let a1 be TopStruct;
  cluster empty compact Element of bool the carrier of a1;
end;

:: BORSUK_6:th 34
theorem
for b1 being TopStruct holds
   {} is empty compact Element of bool the carrier of b1;

:: BORSUK_6:th 35
theorem
for b1 being TopStruct
for b2, b3 being real set
      st b3 < b2
   holds [.b2,b3.] is empty compact Element of bool the carrier of b1;

:: BORSUK_6:th 36
theorem
for b1, b2, b3, b4 being Element of the carrier of I[01] holds
[:[.b1,b2.],[.b3,b4.]:] is compact Element of bool the carrier of [:I[01],I[01]:];

:: BORSUK_6:funcnot 1 => BORSUK_6:func 1
definition
  let a1, a2, a3, a4 be real set;
  func L[01](A1,A2,A3,A4) -> Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(a1,a2),the carrier of Closed-Interval-TSpace(a3,a4) equals
    (L[01]((#)(a3,a4),(a3,a4)(#))) * P[01](a1,a2,(#)(0,1),(0,1)(#));
end;

:: BORSUK_6:def 2
theorem
for b1, b2, b3, b4 being real set holds
L[01](b1,b2,b3,b4) = (L[01]((#)(b3,b4),(b3,b4)(#))) * P[01](b1,b2,(#)(0,1),(0,1)(#));

:: BORSUK_6:th 37
theorem
for b1, b2, b3, b4 being real set
      st b1 < b2 & b3 < b4
   holds (L[01](b1,b2,b3,b4)) . b1 = b3 & (L[01](b1,b2,b3,b4)) . b2 = b4;

:: BORSUK_6:th 38
theorem
for b1, b2, b3, b4 being real set
      st b1 < b2 & b3 <= b4
   holds L[01](b1,b2,b3,b4) is Function-like quasi_total continuous Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4);

:: BORSUK_6:th 39
theorem
for b1, b2, b3, b4 being real set
   st b1 < b2 & b3 <= b4
for b5 being real set
      st b1 <= b5 & b5 <= b2
   holds (L[01](b1,b2,b3,b4)) . b5 = (((b4 - b3) / (b2 - b1)) * (b5 - b1)) + b3;

:: BORSUK_6:th 40
theorem
for b1, b2 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of I[01]
      st b1 is continuous([:I[01],I[01]:], I[01]) &
         b2 is continuous([:I[01],I[01]:], I[01]) &
         (for b3 being Element of the carrier of [:I[01],I[01]:] holds
            (b1 . b3) * (b2 . b3) is Element of the carrier of I[01])
   holds ex b3 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of I[01] st
      (for b4 being Element of the carrier of [:I[01],I[01]:]
       for b5, b6 being real set
             st b1 . b4 = b5 & b2 . b4 = b6
          holds b3 . b4 = b5 * b6) &
       b3 is continuous([:I[01],I[01]:], I[01]);

:: BORSUK_6:th 41
theorem
for b1, b2 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of I[01]
      st b1 is continuous([:I[01],I[01]:], I[01]) &
         b2 is continuous([:I[01],I[01]:], I[01]) &
         (for b3 being Element of the carrier of [:I[01],I[01]:] holds
            (b1 . b3) + (b2 . b3) is Element of the carrier of I[01])
   holds ex b3 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of I[01] st
      (for b4 being Element of the carrier of [:I[01],I[01]:]
       for b5, b6 being real set
             st b1 . b4 = b5 & b2 . b4 = b6
          holds b3 . b4 = b5 + b6) &
       b3 is continuous([:I[01],I[01]:], I[01]);

:: BORSUK_6:th 42
theorem
for b1, b2 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of I[01]
      st b1 is continuous([:I[01],I[01]:], I[01]) &
         b2 is continuous([:I[01],I[01]:], I[01]) &
         (for b3 being Element of the carrier of [:I[01],I[01]:] holds
            (b1 . b3) - (b2 . b3) is Element of the carrier of I[01])
   holds ex b3 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of I[01] st
      (for b4 being Element of the carrier of [:I[01],I[01]:]
       for b5, b6 being real set
             st b1 . b4 = b5 & b2 . b4 = b6
          holds b3 . b4 = b5 - b6) &
       b3 is continuous([:I[01],I[01]:], I[01]);

:: BORSUK_6:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
      st b4 is continuous(I[01], b1)
   holds b4 * L[01]((0,1)(#),(#)(0,1)) is Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of b1;

:: BORSUK_6:th 44
theorem
for b1 being non empty TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
      st b4 . 0 = b2 & b4 . 1 = b3
   holds (b4 * L[01]((0,1)(#),(#)(0,1))) . 0 = b3 &
    (b4 * L[01]((0,1)(#),(#)(0,1))) . 1 = b2;

:: BORSUK_6:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
      st b4 is continuous(I[01], b1) & b4 . 0 = b2 & b4 . 1 = b3
   holds - b4 is continuous(I[01], b1) & (- b4) . 0 = b3 & (- b4) . 1 = b2;

:: BORSUK_6:prednot 1 => BORSUK_6:pred 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of the carrier of a1;
  redefine pred a2,a3 are_connected;
  symmetry;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2, a3 being Element of the carrier of a1
::        st a2,a3 are_connected
::     holds a3,a2 are_connected;
  reflexivity;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2 being Element of the carrier of a1 holds
::     a2,a2 are_connected;
end;

:: BORSUK_6:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3 are_connected & b3,b4 are_connected
   holds b2,b4 are_connected;

:: BORSUK_6:th 50
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
   st b2,b3 are_connected
for b4 being Path of b2,b3 holds
   b4 = - - b4;

:: BORSUK_6:th 51
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3 holds
   b4 = - - b4;

:: BORSUK_6:funcnot 2 => BORSUK_2:func 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  let a5 be Path of a2,a3;
  let a6 be Path of a3,a4;
  func A5 + A6 -> Path of a2,a4 means
    for b1 being Element of the carrier of I[01] holds
       (b1 <= 1 / 2 implies it . b1 = a5 . (2 * b1)) &
        (1 / 2 <= b1 implies it . b1 = a6 . ((2 * b1) - 1));
end;

:: BORSUK_6:def 4
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b2,b3
for b6 being Path of b3,b4
for b7 being Path of b2,b4 holds
      b7 = b5 + b6
   iff
      for b8 being Element of the carrier of I[01] holds
         (b8 <= 1 / 2 implies b7 . b8 = b5 . (2 * b8)) &
          (1 / 2 <= b8 implies b7 . b8 = b6 . ((2 * b8) - 1));

:: BORSUK_6:funcnot 3 => BORSUK_2:func 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Path of a2,a3;
  func - A4 -> Path of a3,a2 means
    for b1 being Element of the carrier of I[01] holds
       it . b1 = a4 . (1 - b1);
end;

:: BORSUK_6:def 5
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being Path of b3,b2 holds
      b5 = - b4
   iff
      for b6 being Element of the carrier of I[01] holds
         b5 . b6 = b4 . (1 - b6);

:: BORSUK_6:funcnot 4 => BORSUK_6:func 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Path of a2,a3;
  let a5 be Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of I[01];
  assume a5 . 0 = 0 & a5 . 1 = 1 & a2,a3 are_connected;
  func RePar(A4,A5) -> Path of a2,a3 equals
    a4 * a5;
end;

:: BORSUK_6:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of I[01]
      st b5 . 0 = 0 & b5 . 1 = 1 & b2,b3 are_connected
   holds RePar(b4,b5) = b4 * b5;

:: BORSUK_6:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of I[01]
      st b5 . 0 = 0 & b5 . 1 = 1 & b2,b3 are_connected
   holds RePar(b4,b5),b4 are_homotopic;

:: BORSUK_6:th 54
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of I[01]
      st b5 . 0 = 0 & b5 . 1 = 1
   holds RePar(b4,b5),b4 are_homotopic;

:: BORSUK_6:funcnot 5 => BORSUK_6:func 3
definition
  func 1RP -> Function-like quasi_total Relation of the carrier of I[01],the carrier of I[01] means
    for b1 being Element of the carrier of I[01] holds
       (b1 <= 1 / 2 implies it . b1 = 2 * b1) &
        (b1 <= 1 / 2 or it . b1 = 1);
end;

:: BORSUK_6:def 7
theorem
for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of I[01] holds
      b1 = 1RP
   iff
      for b2 being Element of the carrier of I[01] holds
         (b2 <= 1 / 2 implies b1 . b2 = 2 * b2) &
          (b2 <= 1 / 2 or b1 . b2 = 1);

:: BORSUK_6:funcreg 7
registration
  cluster 1RP -> Function-like quasi_total continuous;
end;

:: BORSUK_6:th 55
theorem
1RP . 0 = 0 & 1RP . 1 = 1;

:: BORSUK_6:funcnot 6 => BORSUK_6:func 4
definition
  func 2RP -> Function-like quasi_total Relation of the carrier of I[01],the carrier of I[01] means
    for b1 being Element of the carrier of I[01] holds
       (b1 <= 1 / 2 implies it . b1 = 0) &
        (b1 <= 1 / 2 or it . b1 = (2 * b1) - 1);
end;

:: BORSUK_6:def 8
theorem
for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of I[01] holds
      b1 = 2RP
   iff
      for b2 being Element of the carrier of I[01] holds
         (b2 <= 1 / 2 implies b1 . b2 = 0) &
          (b2 <= 1 / 2 or b1 . b2 = (2 * b2) - 1);

:: BORSUK_6:funcreg 8
registration
  cluster 2RP -> Function-like quasi_total continuous;
end;

:: BORSUK_6:th 56
theorem
2RP . 0 = 0 & 2RP . 1 = 1;

:: BORSUK_6:funcnot 7 => BORSUK_6:func 5
definition
  func 3RP -> Function-like quasi_total Relation of the carrier of I[01],the carrier of I[01] means
    for b1 being Element of the carrier of I[01] holds
       (b1 <= 1 / 2 implies it . b1 = (1 / 2) * b1) &
        (1 / 2 < b1 & b1 <= 3 / 4 implies it . b1 = b1 - (1 / 4)) &
        (b1 <= 3 / 4 or it . b1 = (2 * b1) - 1);
end;

:: BORSUK_6:def 9
theorem
for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of I[01] holds
      b1 = 3RP
   iff
      for b2 being Element of the carrier of I[01] holds
         (b2 <= 1 / 2 implies b1 . b2 = (1 / 2) * b2) &
          (1 / 2 < b2 & b2 <= 3 / 4 implies b1 . b2 = b2 - (1 / 4)) &
          (b2 <= 3 / 4 or b1 . b2 = (2 * b2) - 1);

:: BORSUK_6:funcreg 9
registration
  cluster 3RP -> Function-like quasi_total continuous;
end;

:: BORSUK_6:th 57
theorem
3RP . 0 = 0 & 3RP . 1 = 1;

:: BORSUK_6:th 58
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b3,b3
      st b2,b3 are_connected
   holds RePar(b4,1RP) = b4 + b5;

:: BORSUK_6:th 59
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b2,b2
      st b2,b3 are_connected
   holds RePar(b4,2RP) = b5 + b4;

:: BORSUK_6:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b3,b4
for b8 being Path of b4,b5
      st b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected
   holds RePar((b6 + b7) + b8,3RP) = b6 + (b7 + b8);

:: BORSUK_6:funcnot 8 => BORSUK_6:func 6
definition
  func LowerLeftUnitTriangle -> Element of bool the carrier of [:I[01],I[01]:] means
    for b1 being set holds
          b1 in it
       iff
          ex b2, b3 being Element of the carrier of I[01] st
             b1 = [b2,b3] & b3 <= 1 - (2 * b2);
end;

:: BORSUK_6:def 10
theorem
for b1 being Element of bool the carrier of [:I[01],I[01]:] holds
      b1 = LowerLeftUnitTriangle
   iff
      for b2 being set holds
            b2 in b1
         iff
            ex b3, b4 being Element of the carrier of I[01] st
               b2 = [b3,b4] & b4 <= 1 - (2 * b3);

:: BORSUK_6:funcnot 9 => BORSUK_6:func 6
notation
  synonym IAA for LowerLeftUnitTriangle;
end;

:: BORSUK_6:funcnot 10 => BORSUK_6:func 7
definition
  func UpperUnitTriangle -> Element of bool the carrier of [:I[01],I[01]:] means
    for b1 being set holds
          b1 in it
       iff
          ex b2, b3 being Element of the carrier of I[01] st
             b1 = [b2,b3] & 1 - (2 * b2) <= b3 & (2 * b2) - 1 <= b3;
end;

:: BORSUK_6:def 11
theorem
for b1 being Element of bool the carrier of [:I[01],I[01]:] holds
      b1 = UpperUnitTriangle
   iff
      for b2 being set holds
            b2 in b1
         iff
            ex b3, b4 being Element of the carrier of I[01] st
               b2 = [b3,b4] & 1 - (2 * b3) <= b4 & (2 * b3) - 1 <= b4;

:: BORSUK_6:funcnot 11 => BORSUK_6:func 7
notation
  synonym IBB for UpperUnitTriangle;
end;

:: BORSUK_6:funcnot 12 => BORSUK_6:func 8
definition
  func LowerRightUnitTriangle -> Element of bool the carrier of [:I[01],I[01]:] means
    for b1 being set holds
          b1 in it
       iff
          ex b2, b3 being Element of the carrier of I[01] st
             b1 = [b2,b3] & b3 <= (2 * b2) - 1;
end;

:: BORSUK_6:def 12
theorem
for b1 being Element of bool the carrier of [:I[01],I[01]:] holds
      b1 = LowerRightUnitTriangle
   iff
      for b2 being set holds
            b2 in b1
         iff
            ex b3, b4 being Element of the carrier of I[01] st
               b2 = [b3,b4] & b4 <= (2 * b3) - 1;

:: BORSUK_6:funcnot 13 => BORSUK_6:func 8
notation
  synonym ICC for LowerRightUnitTriangle;
end;

:: BORSUK_6:th 61
theorem
LowerLeftUnitTriangle = {b1 where b1 is Element of the carrier of [:I[01],I[01]:]: b1 `2 <= 1 - (2 * (b1 `1))};

:: BORSUK_6:th 62
theorem
UpperUnitTriangle = {b1 where b1 is Element of the carrier of [:I[01],I[01]:]: 1 - (2 * (b1 `1)) <= b1 `2 &
 (2 * (b1 `1)) - 1 <= b1 `2};

:: BORSUK_6:th 63
theorem
LowerRightUnitTriangle = {b1 where b1 is Element of the carrier of [:I[01],I[01]:]: b1 `2 <= (2 * (b1 `1)) - 1};

:: BORSUK_6:funcreg 10
registration
  cluster LowerLeftUnitTriangle -> non empty closed;
end;

:: BORSUK_6:funcreg 11
registration
  cluster UpperUnitTriangle -> non empty closed;
end;

:: BORSUK_6:funcreg 12
registration
  cluster LowerRightUnitTriangle -> non empty closed;
end;

:: BORSUK_6:th 64
theorem
(LowerLeftUnitTriangle \/ UpperUnitTriangle) \/ LowerRightUnitTriangle = [:[.0,1.],[.0,1.]:];

:: BORSUK_6:th 65
theorem
LowerLeftUnitTriangle /\ UpperUnitTriangle = {b1 where b1 is Element of the carrier of [:I[01],I[01]:]: b1 `2 = 1 - (2 * (b1 `1))};

:: BORSUK_6:th 66
theorem
LowerRightUnitTriangle /\ UpperUnitTriangle = {b1 where b1 is Element of the carrier of [:I[01],I[01]:]: b1 `2 = (2 * (b1 `1)) - 1};

:: BORSUK_6:th 67
theorem
for b1 being Element of the carrier of [:I[01],I[01]:]
      st b1 in LowerLeftUnitTriangle
   holds b1 `1 <= 1 / 2;

:: BORSUK_6:th 68
theorem
for b1 being Element of the carrier of [:I[01],I[01]:]
      st b1 in LowerRightUnitTriangle
   holds 1 / 2 <= b1 `1;

:: BORSUK_6:th 69
theorem
for b1 being Element of the carrier of I[01] holds
   [0,b1] in LowerLeftUnitTriangle;

:: BORSUK_6:th 70
theorem
for b1 being set
      st [0,b1] in UpperUnitTriangle
   holds b1 = 1;

:: BORSUK_6:th 71
theorem
for b1 being set
      st [b1,1] in LowerRightUnitTriangle
   holds b1 = 1;

:: BORSUK_6:th 72
theorem
[0,1] in UpperUnitTriangle;

:: BORSUK_6:th 73
theorem
for b1 being Element of the carrier of I[01] holds
   [b1,1] in UpperUnitTriangle;

:: BORSUK_6:th 74
theorem
[1 / 2,0] in LowerRightUnitTriangle &
 [1,1] in LowerRightUnitTriangle;

:: BORSUK_6:th 75
theorem
[1 / 2,0] in UpperUnitTriangle;

:: BORSUK_6:th 76
theorem
for b1 being Element of the carrier of I[01] holds
   [1,b1] in LowerRightUnitTriangle;

:: BORSUK_6:th 77
theorem
for b1 being Element of the carrier of I[01]
      st 1 / 2 <= b1
   holds [b1,0] in LowerRightUnitTriangle;

:: BORSUK_6:th 78
theorem
for b1 being Element of the carrier of I[01]
      st b1 <= 1 / 2
   holds [b1,0] in LowerLeftUnitTriangle;

:: BORSUK_6:th 79
theorem
for b1 being Element of the carrier of I[01]
      st b1 < 1 / 2
   holds not [b1,0] in UpperUnitTriangle & not [b1,0] in LowerRightUnitTriangle;

:: BORSUK_6:th 80
theorem
LowerLeftUnitTriangle /\ LowerRightUnitTriangle = {[1 / 2,0]};

:: BORSUK_6:th 81
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b3,b4
for b8 being Path of b4,b5
      st b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected
   holds (b6 + b7) + b8,b6 + (b7 + b8) are_homotopic;

:: BORSUK_6:th 82
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b3,b4
for b8 being Path of b4,b5 holds
   (b6 + b7) + b8,b6 + (b7 + b8) are_homotopic;

:: BORSUK_6:th 83
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7, b8 being Path of b3,b4
      st b2,b3 are_connected & b3,b4 are_connected & b5,b6 are_homotopic & b7,b8 are_homotopic
   holds b5 + b7,b6 + b8 are_homotopic;

:: BORSUK_6:th 84
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7, b8 being Path of b3,b4
      st b5,b6 are_homotopic & b7,b8 are_homotopic
   holds b5 + b7,b6 + b8 are_homotopic;

:: BORSUK_6:th 85
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Path of b2,b3
      st b2,b3 are_connected & b4,b5 are_homotopic
   holds - b4,- b5 are_homotopic;

:: BORSUK_6:th 86
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Path of b2,b3
      st b4,b5 are_homotopic
   holds - b4,- b5 are_homotopic;

:: BORSUK_6:th 87
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5, b6 being Path of b2,b3
      st b4,b5 are_homotopic & b5,b6 are_homotopic
   holds b4,b6 are_homotopic;

:: BORSUK_6:th 88
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b3,b3
      st b2,b3 are_connected
   holds b4 + b5,b4 are_homotopic;

:: BORSUK_6:th 89
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b3,b3 holds
   b4 + b5,b4 are_homotopic;

:: BORSUK_6:th 90
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b2,b2
      st b2,b3 are_connected
   holds b5 + b4,b4 are_homotopic;

:: BORSUK_6:th 91
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b2,b2 holds
   b5 + b4,b4 are_homotopic;

:: BORSUK_6:th 92
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b2,b2
      st b2,b3 are_connected
   holds b4 + - b4,b5 are_homotopic;

:: BORSUK_6:th 93
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b2,b2 holds
   b4 + - b4,b5 are_homotopic;

:: BORSUK_6:th 94
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b3,b3
      st b2,b3 are_connected
   holds (- b4) + b4,b5 are_homotopic;

:: BORSUK_6:th 95
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
for b5 being constant Path of b3,b3 holds
   (- b4) + b4,b5 are_homotopic;

:: BORSUK_6:th 96
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being constant Path of b2,b2 holds
b3,b4 are_homotopic;

:: BORSUK_6:modenot 1 => BORSUK_6:mode 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4, a5 be Path of a2,a3;
  assume a4,a5 are_homotopic;
  mode Homotopy of A4,A5 -> Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of a1 means
    it is continuous([:I[01],I[01]:], a1) &
     (for b1 being Element of the carrier of I[01] holds
        it .(b1,0) = a4 . b1 & it .(b1,1) = a5 . b1 & it .(0,b1) = a2 & it .(1,b1) = a3);
end;

:: BORSUK_6:dfs 11
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4, a5 be Path of a2,a3;
  let a6 be Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of a1;
To prove
     a6 is Homotopy of a4,a5
it is sufficient to prove
thus a4,a5 are_homotopic;
  thus a6 is continuous([:I[01],I[01]:], a1) &
     (for b1 being Element of the carrier of I[01] holds
        a6 .(b1,0) = a4 . b1 & a6 .(b1,1) = a5 . b1 & a6 .(0,b1) = a2 & a6 .(1,b1) = a3);

:: BORSUK_6:def 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Path of b2,b3
   st b4,b5 are_homotopic
for b6 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of b1 holds
      b6 is Homotopy of b4,b5
   iff
      b6 is continuous([:I[01],I[01]:], b1) &
       (for b7 being Element of the carrier of I[01] holds
          b6 .(b7,0) = b4 . b7 & b6 .(b7,1) = b5 . b7 & b6 .(0,b7) = b2 & b6 .(1,b7) = b3);