Article BORSUK_1, MML version 4.99.1005

:: BORSUK_1:th 5
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
      st b1 c= b3 " b2
   holds b3 .: b1 c= b2;

:: BORSUK_1:th 6
theorem
for b1, b2, b3 being set holds
(b1 --> b2) .: b3 c= {b2};

:: BORSUK_1:th 9
theorem
for b1, b2, b3 being set
      st b1 c= [:b2,b3:]
   holds (.: pr1(b2,b3)) . b1 = (pr1(b2,b3)) .: b1;

:: BORSUK_1:th 10
theorem
for b1, b2, b3 being set
      st b1 c= [:b2,b3:]
   holds (.: pr2(b2,b3)) . b1 = (pr2(b2,b3)) .: b1;

:: BORSUK_1:th 12
theorem
for b1, b2 being set
for b3 being Element of bool b1
for b4 being Element of bool b2
      st [:b3,b4:] <> {}
   holds (pr1(b1,b2)) .: [:b3,b4:] = b3 & (pr2(b1,b2)) .: [:b3,b4:] = b4;

:: BORSUK_1:th 13
theorem
for b1, b2 being set
for b3 being Element of bool b1
for b4 being Element of bool b2
      st [:b3,b4:] <> {}
   holds (.: pr1(b1,b2)) . [:b3,b4:] = b3 &
    (.: pr2(b1,b2)) . [:b3,b4:] = b4;

:: BORSUK_1:th 14
theorem
for b1, b2 being set
for b3 being Element of bool [:b1,b2:]
for b4 being Element of bool bool [:b1,b2:]
      st for b5 being set
              st b5 in b4
           holds b5 c= b3 &
            (ex b6 being Element of bool b1 st
               ex b7 being Element of bool b2 st
                  b5 = [:b6,b7:])
   holds [:union ((.: pr1(b1,b2)) .: b4),meet ((.: pr2(b1,b2)) .: b4):] c= b3;

:: BORSUK_1:th 15
theorem
for b1, b2 being set
for b3 being Element of bool [:b1,b2:]
for b4 being Element of bool bool [:b1,b2:]
      st for b5 being set
              st b5 in b4
           holds b5 c= b3 &
            (ex b6 being Element of bool b1 st
               ex b7 being Element of bool b2 st
                  b5 = [:b6,b7:])
   holds [:meet ((.: pr1(b1,b2)) .: b4),union ((.: pr2(b1,b2)) .: b4):] c= b3;

:: BORSUK_1:th 16
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool bool b1 holds
   union ((.: b3) .: b4) = b3 .: union b4;

:: BORSUK_1:th 17
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
   union union b2 = union {union b3 where b3 is Element of bool b1: b3 in b2};

:: BORSUK_1:th 18
theorem
for b1 being set
for b2 being Element of bool bool b1
   st union b2 = b1
for b3 being Element of bool b2
for b4 being Element of bool b1
      st b4 = union b3
   holds b4 ` c= union (b3 `);

:: BORSUK_1:th 19
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
      st for b6, b7 being Element of b1
              st b4 . b6 = b4 . b7
           holds b5 . b6 = b5 . b7
   holds ex b6 being Function-like quasi_total Relation of b2,b3 st
      b6 * b4 = b5;

:: BORSUK_1:th 20
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like quasi_total Relation of b1,b2
for b6 being Function-like quasi_total Relation of b2,b3 holds
   b5 " {b4} c= (b6 * b5) " {b6 . b4};

:: BORSUK_1:th 21
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Element of b1
for b6 being Element of b3 holds
   [:b4,id b3:] .(b5,b6) = [b4 . b5,b6];

:: BORSUK_1:th 23
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Element of bool b1
for b6 being Element of bool b3 holds
   [:b4,id b3:] .: [:b5,b6:] = [:b4 .: b5,b6:];

:: BORSUK_1:th 24
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Element of b2
for b6 being Element of b3 holds
   [:b4,id b3:] " {[b5,b6]} = [:b4 " {b5},{b6}:];

:: BORSUK_1:th 25
theorem
for b1 being non empty set
for b2 being Element of bool bool b1
for b3 being Element of bool b2 holds
   union b3 is Element of bool b1;

:: BORSUK_1:th 26
theorem
for b1 being set
for b2 being a_partition of b1
for b3, b4 being Element of bool b2 holds
union (b3 /\ b4) = (union b3) /\ union b4;

:: BORSUK_1:th 27
theorem
for b1 being non empty set
for b2 being a_partition of b1
for b3 being Element of bool b2
for b4 being Element of bool b1
      st b4 = union b3
   holds b4 ` = union (b3 `);

:: BORSUK_1:th 28
theorem
for b1 being non empty set
for b2 being symmetric transitive total Relation of b1,b1 holds
   Class b2 is not empty;

:: BORSUK_1:exreg 1
registration
  let a1 be non empty set;
  cluster non empty with_non-empty_elements a_partition of a1;
end;

:: BORSUK_1:funcnot 1 => BORSUK_1:func 1
definition
  let a1 be non empty set;
  let a2 be non empty a_partition of a1;
  func proj A2 -> Function-like quasi_total Relation of a1,a2 means
    for b1 being Element of a1 holds
       b1 in it . b1;
end;

:: BORSUK_1:def 1
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 = proj b2
   iff
      for b4 being Element of b1 holds
         b4 in b3 . b4;

:: BORSUK_1:th 29
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of b1
for b4 being Element of b2
      st b3 in b4
   holds b4 = (proj b2) . b3;

:: BORSUK_1:th 30
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of b2 holds
   b3 = (proj b2) " {b3};

:: BORSUK_1:th 31
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of bool b2 holds
   (proj b2) " b3 = union b3;

:: BORSUK_1:th 32
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of b2 holds
   ex b4 being Element of b1 st
      (proj b2) . b4 = b3;

:: BORSUK_1:th 33
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of bool b1
      st for b4 being Element of bool b1
              st b4 in b2 & b4 meets b3
           holds b4 c= b3
   holds b3 = (proj b2) " ((proj b2) .: b3);

:: BORSUK_1:th 35
theorem
for b1 being TopStruct
for b2 being SubSpace of b1 holds
   the carrier of b2 c= the carrier of b1;

:: BORSUK_1:funcnot 2 => BORSUK_1:func 2
definition
  let a1 be 1-sorted;
  let a2 be non empty 1-sorted;
  let a3 be Element of the carrier of a2;
  func A1 --> A3 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 equals
    (the carrier of a1) --> a3;
end;

:: BORSUK_1:def 2
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Element of the carrier of b2 holds
   b1 --> b3 = (the carrier of b1) --> b3;

:: BORSUK_1:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b2 holds
   b1 --> b3 is continuous(b1, b2);

:: BORSUK_1:funcreg 1
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like TopStruct;
  let a3 be Element of the carrier of a2;
  cluster a1 --> a3 -> Function-like quasi_total continuous;
end;

:: BORSUK_1:attrnot 1 => PRE_TOPC:attr 5
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 continuous means
    for b1 being Element of the carrier of a1
    for b2 being a_neighborhood of a3 . b1 holds
       ex b3 being a_neighborhood of b1 st
          a3 .: b3 c= b2;
end;

:: BORSUK_1:dfs 3
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is continuous
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
    for b2 being a_neighborhood of a3 . b1 holds
       ex b3 being a_neighborhood of b1 st
          a3 .: b3 c= b2;

:: BORSUK_1:def 3
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 holds
      b3 is continuous(b1, b2)
   iff
      for b4 being Element of the carrier of b1
      for b5 being a_neighborhood of b3 . b4 holds
         ex b6 being a_neighborhood of b4 st
            b3 .: b6 c= b5;

:: BORSUK_1:exreg 2
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like TopStruct;
  cluster Relation-like Function-like quasi_total total continuous Relation of the carrier of a1,the carrier of a2;
end;

:: BORSUK_1:funcreg 2
registration
  let a1, a2, a3 be non empty TopSpace-like TopStruct;
  let a4 be Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2;
  let a5 be Function-like quasi_total continuous Relation of the carrier of a2,the carrier of a3;
  cluster a4 * a5 -> Function-like quasi_total continuous;
end;

:: BORSUK_1:th 37
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
for b4 being Element of bool the carrier of b2 holds
   b3 " Int b4 c= Int (b3 " b4);

:: BORSUK_1:th 38
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1
for b5 being a_neighborhood of b3 holds
   b4 " b5 is a_neighborhood of b4 " {b3};

:: BORSUK_1:funcnot 3 => BORSUK_1:func 3
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Element of the carrier of a2;
  let a4 be Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2;
  let a5 be a_neighborhood of a3;
  redefine func a4 " a5 -> a_neighborhood of a4 " {a3};
end;

:: BORSUK_1:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being a_neighborhood of b3
      st b2 c= b3
   holds b4 is a_neighborhood of b2;

:: BORSUK_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   {b2} is compact(b1);

:: BORSUK_1:th 42
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds    b3 is compact(b1)
   iff
      b4 is compact(b2);

:: BORSUK_1:funcnot 4 => BORSUK_1:func 4
definition
  let a1, a2 be TopSpace-like TopStruct;
  func [:A1,A2:] -> strict TopSpace-like TopStruct means
    the carrier of it = [:the carrier of a1,the carrier of a2:] &
     the topology of it = {union b1 where b1 is Element of bool bool the carrier of it: b1 c= {[:b2,b3:] where b2 is Element of bool the carrier of a1, b3 is Element of bool the carrier of a2: b2 in the topology of a1 & b3 in the topology of a2}};
end;

:: BORSUK_1:def 5
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being strict TopSpace-like TopStruct holds
      b3 = [:b1,b2:]
   iff
      the carrier of b3 = [:the carrier of b1,the carrier of b2:] &
       the topology of b3 = {union b4 where b4 is Element of bool bool the carrier of b3: b4 c= {[:b5,b6:] where b5 is Element of bool the carrier of b1, b6 is Element of bool the carrier of b2: b5 in the topology of b1 & b6 in the topology of b2}};

:: BORSUK_1:funcreg 3
registration
  let a1, a2 be non empty TopSpace-like TopStruct;
  cluster [:a1,a2:] -> non empty strict TopSpace-like;
end;

:: BORSUK_1:th 45
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
      b3 is open([:b1,b2:])
   iff
      ex b4 being Element of bool bool the carrier of [:b1,b2:] st
         b3 = union b4 &
          (for b5 being set
                st b5 in b4
             holds ex b6 being Element of bool the carrier of b1 st
                ex b7 being Element of bool the carrier of b2 st
                   b5 = [:b6,b7:] & b6 is open(b1) & b7 is open(b2));

:: BORSUK_1:funcnot 5 => BORSUK_1:func 5
definition
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Element of bool the carrier of a1;
  let a4 be Element of bool the carrier of a2;
  redefine func [:a3, a4:] -> Element of bool the carrier of [:a1,a2:];
end;

:: BORSUK_1:funcnot 6 => BORSUK_1:func 6
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Element of the carrier of a1;
  let a4 be Element of the carrier of a2;
  redefine func [a3, a4] -> Element of the carrier of [:a1,a2:];
end;

:: BORSUK_1:th 46
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 is open(b1) & b4 is open(b2)
   holds [:b3,b4:] is open([:b1,b2:]);

:: BORSUK_1:th 47
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
   Int [:b3,b4:] = [:Int b3,Int b4:];

:: BORSUK_1:th 48
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being a_neighborhood of b3
for b6 being a_neighborhood of b4 holds
   [:b5,b6:] is a_neighborhood of [b3,b4];

:: BORSUK_1:th 49
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being a_neighborhood of b3
for b6 being a_neighborhood of b4 holds
   [:b5,b6:] is a_neighborhood of [:b3,b4:];

:: BORSUK_1:funcnot 7 => BORSUK_1:func 7
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Element of the carrier of a1;
  let a4 be Element of the carrier of a2;
  let a5 be a_neighborhood of a3;
  let a6 be a_neighborhood of a4;
  redefine func [:a5, a6:] -> a_neighborhood of [a3,a4];
end;

:: BORSUK_1:th 50
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of [:b1,b2:] holds
   ex b4 being Element of the carrier of b1 st
      ex b5 being Element of the carrier of b2 st
         b3 = [b4,b5];

:: BORSUK_1:funcnot 8 => BORSUK_1:func 8
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Element of bool the carrier of a1;
  let a4 be Element of the carrier of a2;
  let a5 be a_neighborhood of a3;
  let a6 be a_neighborhood of a4;
  redefine func [:a5, a6:] -> a_neighborhood of [:a3,{a4}:];
end;

:: BORSUK_1:funcnot 9 => BORSUK_1:func 9
definition
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Element of bool the carrier of [:a1,a2:];
  func Base-Appr A3 -> Element of bool bool the carrier of [:a1,a2:] equals
    {[:b1,b2:] where b1 is Element of bool the carrier of a1, b2 is Element of bool the carrier of a2: [:b1,b2:] c= a3 & b1 is open(a1) & b2 is open(a2)};
end;

:: BORSUK_1:def 6
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
   Base-Appr b3 = {[:b4,b5:] where b4 is Element of bool the carrier of b1, b5 is Element of bool the carrier of b2: [:b4,b5:] c= b3 & b4 is open(b1) & b5 is open(b2)};

:: BORSUK_1:th 51
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
   Base-Appr b3 is open([:b1,b2:]);

:: BORSUK_1:th 52
theorem
for b1, b2 being TopSpace-like TopStruct
for b3, b4 being Element of bool the carrier of [:b1,b2:]
      st b3 c= b4
   holds Base-Appr b3 c= Base-Appr b4;

:: BORSUK_1:th 53
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
   union Base-Appr b3 c= b3;

:: BORSUK_1:th 54
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
      st b3 is open([:b1,b2:])
   holds b3 = union Base-Appr b3;

:: BORSUK_1:th 55
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
   Int b3 = union Base-Appr b3;

:: BORSUK_1:funcnot 10 => BORSUK_1:func 10
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  func Pr1(A1,A2) -> Function-like quasi_total Relation of bool the carrier of [:a1,a2:],bool the carrier of a1 equals
    .: pr1(the carrier of a1,the carrier of a2);
end;

:: BORSUK_1:def 7
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
Pr1(b1,b2) = .: pr1(the carrier of b1,the carrier of b2);

:: BORSUK_1:funcnot 11 => BORSUK_1:func 11
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  func Pr2(A1,A2) -> Function-like quasi_total Relation of bool the carrier of [:a1,a2:],bool the carrier of a2 equals
    .: pr2(the carrier of a1,the carrier of a2);
end;

:: BORSUK_1:def 8
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
Pr2(b1,b2) = .: pr2(the carrier of b1,the carrier of b2);

:: BORSUK_1:th 56
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
for b4 being Element of bool bool the carrier of [:b1,b2:]
      st for b5 being set
              st b5 in b4
           holds b5 c= b3 &
            (ex b6 being Element of bool the carrier of b1 st
               ex b7 being Element of bool the carrier of b2 st
                  b5 = [:b6,b7:])
   holds [:union ((Pr1(b1,b2)) .: b4),meet ((Pr2(b1,b2)) .: b4):] c= b3;

:: BORSUK_1:th 57
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
for b4 being set
      st b4 in (Pr1(b1,b2)) .: b3
   holds ex b5 being Element of bool the carrier of [:b1,b2:] st
      b5 in b3 &
       b4 = (pr1(the carrier of b1,the carrier of b2)) .: b5;

:: BORSUK_1:th 58
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
for b4 being set
      st b4 in (Pr2(b1,b2)) .: b3
   holds ex b5 being Element of bool the carrier of [:b1,b2:] st
      b5 in b3 &
       b4 = (pr2(the carrier of b1,the carrier of b2)) .: b5;

:: BORSUK_1:th 59
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
   st b3 is open([:b1,b2:])
for b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2 holds
   (b4 = (pr1(the carrier of b1,the carrier of b2)) .: b3 implies b4 is open(b1)) &
    (b5 = (pr2(the carrier of b1,the carrier of b2)) .: b3 implies b5 is open(b2));

:: BORSUK_1:th 60
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
      st b3 is open([:b1,b2:])
   holds (Pr1(b1,b2)) .: b3 is open(b1) & (Pr2(b1,b2)) .: b3 is open(b2);

:: BORSUK_1:th 61
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
      st ((Pr1(b1,b2)) .: b3 = {} or (Pr2(b1,b2)) .: b3 = {})
   holds b3 = {};

:: BORSUK_1:th 62
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
for b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is_a_cover_of [:b4,b5:]
   holds (b5 = {} or (Pr1(b1,b2)) .: b3 is_a_cover_of b4) &
    (b4 = {} or (Pr2(b1,b2)) .: b3 is_a_cover_of b5);

:: BORSUK_1:th 63
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool the carrier of b1
      st b3 is_a_cover_of b4
   holds ex b5 being Element of bool bool the carrier of b1 st
      b5 c= b3 &
       b5 is_a_cover_of b4 &
       (for b6 being set
             st b6 in b5
          holds b6 meets b4);

:: BORSUK_1:th 64
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of [:b1,b2:]
      st b3 is finite & b3 c= (Pr1(b1,b2)) .: b4
   holds ex b5 being Element of bool bool the carrier of [:b1,b2:] st
      b5 c= b4 & b5 is finite & b3 = (Pr1(b1,b2)) .: b5;

:: BORSUK_1:th 65
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st [:b3,b4:] <> {}
   holds (Pr1(b1,b2)) . [:b3,b4:] = b3 & (Pr2(b1,b2)) . [:b3,b4:] = b4;

:: BORSUK_1:th 67
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b2
   st b4 is compact(b2)
for b5 being a_neighborhood of [:b4,{b3}:] holds
   ex b6 being a_neighborhood of b4 st
      ex b7 being a_neighborhood of b3 st
         [:b6,b7:] c= b5;

:: BORSUK_1:funcnot 12 => BORSUK_1:func 12
definition
  let a1 be 1-sorted;
  func TrivDecomp A1 -> a_partition of the carrier of a1 equals
    Class id the carrier of a1;
end;

:: BORSUK_1:def 9
theorem
for b1 being 1-sorted holds
   TrivDecomp b1 = Class id the carrier of b1;

:: BORSUK_1:funcreg 4
registration
  let a1 be non empty 1-sorted;
  cluster TrivDecomp a1 -> non empty;
end;

:: BORSUK_1:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 in TrivDecomp b1
   holds ex b3 being Element of the carrier of b1 st
      b2 = {b3};

:: BORSUK_1:funcnot 13 => BORSUK_1:func 13
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be a_partition of the carrier of a1;
  func space A2 -> strict TopSpace-like TopStruct means
    the carrier of it = a2 &
     the topology of it = {b1 where b1 is Element of bool a2: union b1 in the topology of a1};
end;

:: BORSUK_1:def 10
theorem
for b1 being TopSpace-like TopStruct
for b2 being a_partition of the carrier of b1
for b3 being strict TopSpace-like TopStruct holds
      b3 = space b2
   iff
      the carrier of b3 = b2 &
       the topology of b3 = {b4 where b4 is Element of bool b2: union b4 in the topology of b1};

:: BORSUK_1:funcreg 5
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty a_partition of the carrier of a1;
  cluster space a2 -> non empty strict TopSpace-like;
end;

:: BORSUK_1:th 69
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of bool b2 holds
      union b3 in the topology of b1
   iff
      b3 in the topology of space b2;

:: BORSUK_1:funcnot 14 => BORSUK_1:func 14
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty a_partition of the carrier of a1;
  func Proj A2 -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of space a2 equals
    proj a2;
end;

:: BORSUK_1:def 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1 holds
   Proj b2 = proj b2;

:: BORSUK_1:th 70
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 in (Proj b2) . b3;

:: BORSUK_1:th 71
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of the carrier of space b2 holds
   ex b4 being Element of the carrier of b1 st
      (Proj b2) . b4 = b3;

:: BORSUK_1:th 72
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1 holds
   rng Proj b2 = the carrier of space b2;

:: BORSUK_1:funcnot 15 => BORSUK_1:func 15
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  let a3 be non empty a_partition of the carrier of a2;
  func TrivExt A3 -> non empty a_partition of the carrier of a1 equals
    a3 \/ {{b1} where b1 is Element of the carrier of a1: not b1 in the carrier of a2};
end;

:: BORSUK_1:def 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2 holds
   TrivExt b3 = b3 \/ {{b4} where b4 is Element of the carrier of b1: not b4 in the carrier of b2};

:: BORSUK_1:th 74
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 in TrivExt b3 & not b4 in b3
   holds ex b5 being Element of the carrier of b1 st
      not b5 in [#] b2 & b4 = {b5};

:: BORSUK_1:th 75
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
      st not b4 in the carrier of b2
   holds {b4} in TrivExt b3;

:: BORSUK_1:th 76
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
      st b4 in the carrier of b2
   holds (Proj TrivExt b3) . b4 = (Proj b3) . b4;

:: BORSUK_1:th 77
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
      st not b4 in the carrier of b2
   holds (Proj TrivExt b3) . b4 = {b4};

:: BORSUK_1:th 78
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4, b5 being Element of the carrier of b1
      st not b4 in the carrier of b2 &
         (Proj TrivExt b3) . b4 = (Proj TrivExt b3) . b5
   holds b4 = b5;

:: BORSUK_1:th 79
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
      st (Proj TrivExt b3) . b4 in the carrier of space b3
   holds b4 in the carrier of b2;

:: BORSUK_1:th 80
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being set
      st b4 in the carrier of b2
   holds (Proj TrivExt b3) . b4 in the carrier of space b3;

:: BORSUK_1:modenot 1 => BORSUK_1:mode 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  mode u.s.c._decomposition of A1 -> non empty a_partition of the carrier of a1 means
    for b1 being Element of bool the carrier of a1
       st b1 in it
    for b2 being a_neighborhood of b1 holds
       ex b3 being Element of bool the carrier of a1 st
          b3 is open(a1) &
           b1 c= b3 &
           b3 c= b2 &
           (for b4 being Element of bool the carrier of a1
                 st b4 in it & b4 meets b3
              holds b4 c= b3);
end;

:: BORSUK_1:dfs 12
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty a_partition of the carrier of a1;
To prove
     a2 is u.s.c._decomposition of a1
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
       st b1 in a2
    for b2 being a_neighborhood of b1 holds
       ex b3 being Element of bool the carrier of a1 st
          b3 is open(a1) &
           b1 c= b3 &
           b3 c= b2 &
           (for b4 being Element of bool the carrier of a1
                 st b4 in a2 & b4 meets b3
              holds b4 c= b3);

:: BORSUK_1:def 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1 holds
      b2 is u.s.c._decomposition of b1
   iff
      for b3 being Element of bool the carrier of b1
         st b3 in b2
      for b4 being a_neighborhood of b3 holds
         ex b5 being Element of bool the carrier of b1 st
            b5 is open(b1) &
             b3 c= b5 &
             b5 c= b4 &
             (for b6 being Element of bool the carrier of b1
                   st b6 in b2 & b6 meets b5
                holds b6 c= b5);

:: BORSUK_1:th 81
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being u.s.c._decomposition of b1
for b3 being Element of the carrier of space b2
for b4 being a_neighborhood of (Proj b2) " {b3} holds
   (Proj b2) .: b4 is a_neighborhood of b3;

:: BORSUK_1:th 82
theorem
for b1 being non empty TopSpace-like TopStruct holds
   TrivDecomp b1 is u.s.c._decomposition of b1;

:: BORSUK_1:attrnot 2 => BORSUK_1:attr 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be SubSpace of a1;
  attr a2 is closed means
    for b1 being Element of bool the carrier of a1
          st b1 = the carrier of a2
       holds b1 is closed(a1);
end;

:: BORSUK_1:dfs 13
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be SubSpace of a1;
To prove
     a2 is closed
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 = the carrier of a2
       holds b1 is closed(a1);

:: BORSUK_1:def 14
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
      b2 is closed(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 = the carrier of b2
         holds b3 is closed(b1);

:: BORSUK_1:exreg 3
registration
  let a1 be TopSpace-like TopStruct;
  cluster strict TopSpace-like closed SubSpace of a1;
end;

:: BORSUK_1:exreg 4
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty strict TopSpace-like closed SubSpace of a1;
end;

:: BORSUK_1:funcnot 16 => BORSUK_1:func 16
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty closed SubSpace of a1;
  let a3 be u.s.c._decomposition of a2;
  redefine func TrivExt a3 -> u.s.c._decomposition of a1;
end;

:: BORSUK_1:attrnot 3 => BORSUK_1:attr 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be u.s.c._decomposition of a1;
  attr a2 is DECOMPOSITION-like means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is compact(a1);
end;

:: BORSUK_1:dfs 14
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be u.s.c._decomposition of a1;
To prove
     a2 is DECOMPOSITION-like
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is compact(a1);

:: BORSUK_1:def 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being u.s.c._decomposition of b1 holds
      b2 is DECOMPOSITION-like(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is compact(b1);

:: BORSUK_1:exreg 5
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty with_non-empty_elements DECOMPOSITION-like u.s.c._decomposition of a1;
end;

:: BORSUK_1:modenot 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  mode DECOMPOSITION of a1 is DECOMPOSITION-like u.s.c._decomposition of a1;
end;

:: BORSUK_1:funcnot 17 => BORSUK_1:func 17
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty closed SubSpace of a1;
  let a3 be DECOMPOSITION-like u.s.c._decomposition of a2;
  redefine func TrivExt a3 -> DECOMPOSITION-like u.s.c._decomposition of a1;
end;

:: BORSUK_1:funcnot 18 => BORSUK_1:func 18
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty closed SubSpace of a1;
  let a3 be DECOMPOSITION-like u.s.c._decomposition of a2;
  redefine func space a3 -> strict closed SubSpace of space TrivExt a3;
end;

:: BORSUK_1:funcnot 19 => BORSUK_1:func 19
definition
  func I[01] -> TopStruct means
    for b1 being Element of bool the carrier of TopSpaceMetr RealSpace
          st b1 = [.0,1.]
       holds it = (TopSpaceMetr RealSpace) | b1;
end;

:: BORSUK_1:def 16
theorem
for b1 being TopStruct holds
      b1 = I[01]
   iff
      for b2 being Element of bool the carrier of TopSpaceMetr RealSpace
            st b2 = [.0,1.]
         holds b1 = (TopSpaceMetr RealSpace) | b2;

:: BORSUK_1:funcreg 6
registration
  cluster I[01] -> non empty strict TopSpace-like;
end;

:: BORSUK_1:th 83
theorem
the carrier of I[01] = [.0,1.];

:: BORSUK_1:funcnot 20 => BORSUK_1:func 20
definition
  func 0[01] -> Element of the carrier of I[01] equals
    0;
end;

:: BORSUK_1:def 17
theorem
0[01] = 0;

:: BORSUK_1:funcnot 21 => BORSUK_1:func 21
definition
  func 1[01] -> Element of the carrier of I[01] equals
    1;
end;

:: BORSUK_1:def 18
theorem
1[01] = 1;

:: BORSUK_1:attrnot 4 => BORSUK_1:attr 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is being_a_retraction means
    for b1 being Element of the carrier of a1
          st b1 in the carrier of a2
       holds a3 . b1 = b1;
end;

:: BORSUK_1:dfs 18
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is being_a_retraction
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
          st b1 in the carrier of a2
       holds a3 . b1 = b1;

:: BORSUK_1:def 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_a_retraction(b1, b2)
   iff
      for b4 being Element of the carrier of b1
            st b4 in the carrier of b2
         holds b3 . b4 = b4;

:: BORSUK_1:prednot 1 => BORSUK_1:attr 3
notation
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  synonym a3 is_a_retraction for being_a_retraction;
end;

:: BORSUK_1:prednot 2 => BORSUK_1:pred 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  pred A2 is_a_retract_of A1 means
    ex b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 st
       b1 is being_a_retraction(a1, a2);
end;

:: BORSUK_1:dfs 19
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
To prove
     a2 is_a_retract_of a1
it is sufficient to prove
  thus ex b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 st
       b1 is being_a_retraction(a1, a2);

:: BORSUK_1:def 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
      b2 is_a_retract_of b1
   iff
      ex b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
         b3 is being_a_retraction(b1, b2);

:: BORSUK_1:prednot 3 => BORSUK_1:pred 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  pred A2 is_an_SDR_of A1 means
    ex b1 being Function-like quasi_total continuous Relation of the carrier of [:a1,I[01]:],the carrier of a1 st
       for b2 being Element of the carrier of a1 holds
          b1 . [b2,0[01]] = b2 &
           b1 . [b2,1[01]] in the carrier of a2 &
           (b2 in the carrier of a2 implies for b3 being Element of the carrier of I[01] holds
              b1 . [b2,b3] = b2);
end;

:: BORSUK_1:dfs 20
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
To prove
     a2 is_an_SDR_of a1
it is sufficient to prove
  thus ex b1 being Function-like quasi_total continuous Relation of the carrier of [:a1,I[01]:],the carrier of a1 st
       for b2 being Element of the carrier of a1 holds
          b1 . [b2,0[01]] = b2 &
           b1 . [b2,1[01]] in the carrier of a2 &
           (b2 in the carrier of a2 implies for b3 being Element of the carrier of I[01] holds
              b1 . [b2,b3] = b2);

:: BORSUK_1:def 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
      b2 is_an_SDR_of b1
   iff
      ex b3 being Function-like quasi_total continuous Relation of the carrier of [:b1,I[01]:],the carrier of b1 st
         for b4 being Element of the carrier of b1 holds
            b3 . [b4,0[01]] = b4 &
             b3 . [b4,1[01]] in the carrier of b2 &
             (b4 in the carrier of b2 implies for b5 being Element of the carrier of I[01] holds
                b3 . [b4,b5] = b4);

:: BORSUK_1:th 84
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty closed SubSpace of b1
for b3 being DECOMPOSITION-like u.s.c._decomposition of b2
      st b2 is_a_retract_of b1
   holds space b3 is_a_retract_of space TrivExt b3;

:: BORSUK_1:th 85
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty closed SubSpace of b1
for b3 being DECOMPOSITION-like u.s.c._decomposition of b2
      st b2 is_an_SDR_of b1
   holds space b3 is_an_SDR_of space TrivExt b3;

:: BORSUK_1:th 86
theorem
for b1 being real set holds
      0 <= b1 & b1 <= 1
   iff
      b1 in the carrier of I[01];