Article TOPGRP_1, MML version 4.99.1005

:: TOPGRP_1:exreg 1
registration
  let a1 be set;
  cluster Relation-like Function-like one-to-one quasi_total onto Relation of a1,a1;
end;

:: TOPGRP_1:th 1
theorem
for b1 being 1-sorted holds
   rng id b1 = [#] b1;

:: TOPGRP_1:funcreg 1
registration
  let a1 be 1-sorted;
  cluster (id a1) /" -> Function-like one-to-one quasi_total;
end;

:: TOPGRP_1:th 2
theorem
for b1 being 1-sorted holds
   (id b1) /" = id b1;

:: TOPGRP_1:th 3
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
   (id b1) " b2 = b2;

:: TOPGRP_1:th 4
theorem
for b1 being non empty multMagma
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2 c= b3 & b4 c= b5
   holds b2 * b4 c= b3 * b5;

:: TOPGRP_1:th 5
theorem
for b1 being non empty multMagma
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st b2 c= b3
   holds b2 * b4 c= b3 * b4;

:: TOPGRP_1:th 6
theorem
for b1 being non empty multMagma
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st b2 c= b3
   holds b4 * b2 c= b4 * b3;

:: TOPGRP_1:th 7
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in b2 "
   iff
      b3 " in b2;

:: TOPGRP_1:th 8
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
   b2 " " = b2;

:: TOPGRP_1:th 9
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
   b2 c= b3
iff
   b2 " c= b3 ";

:: TOPGRP_1:th 10
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
   (inverse_op b1) .: b2 = b2 ";

:: TOPGRP_1:th 11
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
   (inverse_op b1) " b2 = b2 ";

:: TOPGRP_1:th 12
theorem
for b1 being non empty Group-like associative multMagma holds
   inverse_op b1 is one-to-one;

:: TOPGRP_1:th 13
theorem
for b1 being non empty Group-like associative multMagma holds
   rng inverse_op b1 = the carrier of b1;

:: TOPGRP_1:funcreg 2
registration
  let a1 be non empty Group-like associative multMagma;
  cluster inverse_op a1 -> Function-like one-to-one quasi_total onto;
end;

:: TOPGRP_1:th 14
theorem
for b1 being non empty Group-like associative multMagma holds
   (inverse_op b1) " = inverse_op b1;

:: TOPGRP_1:th 15
theorem
for b1 being non empty multMagma
for b2, b3 being Element of bool the carrier of b1 holds
(the multF of b1) .: [:b2,b3:] = b2 * b3;

:: TOPGRP_1:funcnot 1 => TOPGRP_1:func 1
definition
  let a1 be non empty multMagma;
  let a2 be Element of the carrier of a1;
  func A2 * -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = a2 * b1;
end;

:: TOPGRP_1:def 1
theorem
for b1 being non empty multMagma
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b3 = b2 *
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = b2 * b4;

:: TOPGRP_1:funcnot 2 => TOPGRP_1:func 2
definition
  let a1 be non empty multMagma;
  let a2 be Element of the carrier of a1;
  func * A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = b1 * a2;
end;

:: TOPGRP_1:def 2
theorem
for b1 being non empty multMagma
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b3 = * b2
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = b4 * b2;

:: TOPGRP_1:funcreg 3
registration
  let a1 be non empty Group-like associative multMagma;
  let a2 be Element of the carrier of a1;
  cluster a2 * -> Function-like one-to-one quasi_total onto;
end;

:: TOPGRP_1:funcreg 4
registration
  let a1 be non empty Group-like associative multMagma;
  let a2 be Element of the carrier of a1;
  cluster * a2 -> Function-like one-to-one quasi_total onto;
end;

:: TOPGRP_1:th 16
theorem
for b1 being non empty multMagma
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 * .: b2 = b3 * b2;

:: TOPGRP_1:th 17
theorem
for b1 being non empty multMagma
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   (* b3) .: b2 = b2 * b3;

:: TOPGRP_1:th 18
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
   b2 * /" = b2 " *;

:: TOPGRP_1:th 19
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
   (* b2) /" = * (b2 ");

:: TOPGRP_1:funcreg 5
registration
  let a1 be TopStruct;
  cluster (id a1) /" -> Function-like quasi_total continuous;
end;

:: TOPGRP_1:th 20
theorem
for b1 being TopStruct holds
   id b1 is being_homeomorphism(b1, b1);

:: TOPGRP_1:condreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  cluster -> non empty (a_neighborhood of a2);
end;

:: TOPGRP_1:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   [#] b1 is a_neighborhood of b2;

:: TOPGRP_1:exreg 2
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  cluster non empty open a_neighborhood of a2;
end;

:: TOPGRP_1:th 22
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 open(b1, b2)
for b4 being Element of the carrier of b1
for b5 being a_neighborhood of b4 holds
   ex b6 being open a_neighborhood of b3 . b4 st
      b6 c= b3 .: b5;

:: TOPGRP_1:th 23
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 for b4 being Element of the carrier of b1
        for b5 being open a_neighborhood of b4 holds
           ex b6 being a_neighborhood of b3 . b4 st
              b6 c= b3 .: b5
   holds b3 is open(b1, b2);

:: TOPGRP_1:th 24
theorem
for b1, b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_homeomorphism(b1, b2)
   iff
      dom b3 = [#] b1 &
       rng b3 = [#] b2 &
       b3 is one-to-one &
       (for b4 being Element of bool the carrier of b2 holds
             b4 is closed(b2)
          iff
             b3 " b4 is closed(b1));

:: TOPGRP_1:th 25
theorem
for b1, b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_homeomorphism(b1, b2)
   iff
      dom b3 = [#] b1 &
       rng b3 = [#] b2 &
       b3 is one-to-one &
       (for b4 being Element of bool the carrier of b1 holds
             b4 is open(b1)
          iff
             b3 .: b4 is open(b2));

:: TOPGRP_1:th 26
theorem
for b1, b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_homeomorphism(b1, b2)
   iff
      dom b3 = [#] b1 &
       rng b3 = [#] b2 &
       b3 is one-to-one &
       (for b4 being Element of bool the carrier of b2 holds
             b4 is open(b2)
          iff
             b3 " b4 is open(b1));

:: TOPGRP_1:th 27
theorem
for b1 being TopSpace-like TopStruct
for 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 bool the carrier of b2 holds
         b3 " Int b4 c= Int (b3 " b4);

:: TOPGRP_1:exreg 3
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty dense Element of bool the carrier of a1;
end;

:: TOPGRP_1:th 28
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 dense Element of bool the carrier of b1
      st b3 is being_homeomorphism(b1, b2)
   holds b3 .: b4 is dense(b2);

:: TOPGRP_1:th 29
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 dense Element of bool the carrier of b2
      st b3 is being_homeomorphism(b1, b2)
   holds b3 " b4 is dense(b1);

:: TOPGRP_1:condreg 2
registration
  let a1, a2 be TopStruct;
  cluster Function-like quasi_total being_homeomorphism -> one-to-one onto continuous (Relation of the carrier of a1,the carrier of a2);
end;

:: TOPGRP_1:condreg 3
registration
  let a1, a2 be non empty TopStruct;
  cluster Function-like quasi_total being_homeomorphism -> open (Relation of the carrier of a1,the carrier of a2);
end;

:: TOPGRP_1:exreg 4
registration
  let a1 be TopStruct;
  cluster Relation-like Function-like quasi_total being_homeomorphism Relation of the carrier of a1,the carrier of a1;
end;

:: TOPGRP_1:funcreg 6
registration
  let a1 be TopStruct;
  let a2 be Function-like quasi_total being_homeomorphism Relation of the carrier of a1,the carrier of a1;
  cluster a2 /" -> Function-like quasi_total being_homeomorphism;
end;

:: TOPGRP_1:modenot 1 => TOPGRP_1:mode 1
definition
  let a1, a2 be TopStruct;
  assume a1,a2 are_homeomorphic;
  mode Homeomorphism of A1,A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
    it is being_homeomorphism(a1, a2);
end;

:: TOPGRP_1:dfs 3
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 Homeomorphism of a1,a2
it is sufficient to prove
thus a1,a2 are_homeomorphic;
  thus a3 is being_homeomorphism(a1, a2);

:: TOPGRP_1:def 3
theorem
for b1, b2 being TopStruct
   st b1,b2 are_homeomorphic
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is Homeomorphism of b1,b2
   iff
      b3 is being_homeomorphism(b1, b2);

:: TOPGRP_1:modenot 2 => TOPGRP_1:mode 2
definition
  let a1 be TopStruct;
  mode Homeomorphism of A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    it is being_homeomorphism(a1, a1);
end;

:: TOPGRP_1:dfs 4
definiens
  let a1 be TopStruct;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is Homeomorphism of a1
it is sufficient to prove
  thus a2 is being_homeomorphism(a1, a1);

:: TOPGRP_1:def 4
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b2 is Homeomorphism of b1
   iff
      b2 is being_homeomorphism(b1, b1);

:: TOPGRP_1:modenot 3 => TOPGRP_1:mode 3
definition
  let a1 be TopStruct;
  redefine mode Homeomorphism of a1 -> Homeomorphism of a1,a1;
end;

:: TOPGRP_1:funcnot 3 => TOPGRP_1:func 3
definition
  let a1 be TopStruct;
  redefine func id a1 -> Homeomorphism of a1,a1;
end;

:: TOPGRP_1:funcnot 4 => TOPGRP_1:func 4
definition
  let a1 be TopStruct;
  redefine func id a1 -> Homeomorphism of a1;
end;

:: TOPGRP_1:condreg 4
registration
  let a1 be TopStruct;
  cluster -> being_homeomorphism (Homeomorphism of a1);
end;

:: TOPGRP_1:th 30
theorem
for b1 being TopStruct
for b2 being Homeomorphism of b1 holds
   b2 /" is Homeomorphism of b1;

:: TOPGRP_1:th 31
theorem
for b1 being TopStruct
for b2, b3 being Homeomorphism of b1 holds
b2 * b3 is Homeomorphism of b1;

:: TOPGRP_1:funcnot 5 => TOPGRP_1:func 5
definition
  let a1 be TopStruct;
  func HomeoGroup A1 -> strict multMagma means
    for b1 being set holds
       (b1 in the carrier of it implies b1 is Homeomorphism of a1) &
        (b1 is Homeomorphism of a1 implies b1 in the carrier of it) &
        (for b2, b3 being Homeomorphism of a1 holds
        (the multF of it) .(b2,b3) = b3 * b2);
end;

:: TOPGRP_1:def 5
theorem
for b1 being TopStruct
for b2 being strict multMagma holds
      b2 = HomeoGroup b1
   iff
      for b3 being set holds
         (b3 in the carrier of b2 implies b3 is Homeomorphism of b1) &
          (b3 is Homeomorphism of b1 implies b3 in the carrier of b2) &
          (for b4, b5 being Homeomorphism of b1 holds
          (the multF of b2) .(b4,b5) = b5 * b4);

:: TOPGRP_1:funcreg 7
registration
  let a1 be TopStruct;
  cluster HomeoGroup a1 -> non empty strict;
end;

:: TOPGRP_1:th 32
theorem
for b1 being TopStruct
for b2, b3 being Homeomorphism of b1
for b4, b5 being Element of the carrier of HomeoGroup b1
      st b2 = b4 & b3 = b5
   holds b4 * b5 = b3 * b2;

:: TOPGRP_1:funcreg 8
registration
  let a1 be TopStruct;
  cluster HomeoGroup a1 -> strict Group-like associative;
end;

:: TOPGRP_1:th 33
theorem
for b1 being TopStruct holds
   id b1 = 1_ HomeoGroup b1;

:: TOPGRP_1:th 34
theorem
for b1 being TopStruct
for b2 being Homeomorphism of b1
for b3 being Element of the carrier of HomeoGroup b1
      st b2 = b3
   holds b3 " = b2 /";

:: TOPGRP_1:attrnot 1 => TOPGRP_1:attr 1
definition
  let a1 be TopStruct;
  attr a1 is homogeneous means
    for b1, b2 being Element of the carrier of a1 holds
    ex b3 being Homeomorphism of a1 st
       b3 . b1 = b2;
end;

:: TOPGRP_1:dfs 6
definiens
  let a1 be TopStruct;
To prove
     a1 is homogeneous
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1 holds
    ex b3 being Homeomorphism of a1 st
       b3 . b1 = b2;

:: TOPGRP_1:def 6
theorem
for b1 being TopStruct holds
      b1 is homogeneous
   iff
      for b2, b3 being Element of the carrier of b1 holds
      ex b4 being Homeomorphism of b1 st
         b4 . b2 = b3;

:: TOPGRP_1:condreg 5
registration
  cluster non empty trivial -> homogeneous (TopStruct);
end;

:: TOPGRP_1:exreg 5
registration
  cluster non empty trivial strict TopSpace-like TopStruct;
end;

:: TOPGRP_1:th 35
theorem
for b1 being non empty TopSpace-like homogeneous TopStruct
      st ex b2 being Element of the carrier of b1 st
           {b2} is closed(b1)
   holds b1 is being_T1;

:: TOPGRP_1:th 36
theorem
for b1 being non empty TopSpace-like homogeneous TopStruct
      st ex b2 being Element of the carrier of b1 st
           for b3 being Element of bool the carrier of b1
                 st b3 is open(b1) & b2 in b3
              holds ex b4 being Element of bool the carrier of b1 st
                 b2 in b4 & b4 is open(b1) & Cl b4 c= b3
   holds b1 is being_T3;

:: TOPGRP_1:structnot 1 => TOPGRP_1:struct 1
definition
  struct(multMagmaTopStruct) TopGrStr(#
    carrier -> set,
    multF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
    topology -> Element of bool bool the carrier of it
  #);
end;

:: TOPGRP_1:attrnot 2 => TOPGRP_1:attr 2
definition
  let a1 be TopGrStr;
  attr a1 is strict;
end;

:: TOPGRP_1:exreg 6
registration
  cluster strict TopGrStr;
end;

:: TOPGRP_1:aggrnot 1 => TOPGRP_1:aggr 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of bool bool a1;
  aggr TopGrStr(#a1,a2,a3#) -> strict TopGrStr;
end;

:: TOPGRP_1:funcreg 9
registration
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of bool bool a1;
  cluster TopGrStr(#a1,a2,a3#) -> non empty strict;
end;

:: TOPGRP_1:funcreg 10
registration
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:{a1},{a1}:],{a1};
  let a3 be Element of bool bool {a1};
  cluster TopGrStr(#{a1},a2,a3#) -> trivial strict;
end;

:: TOPGRP_1:condreg 6
registration
  cluster non empty trivial -> Group-like associative commutative (multMagma);
end;

:: TOPGRP_1:funcreg 11
registration
  let a1 be set;
  cluster 1TopSp {a1} -> trivial;
end;

:: TOPGRP_1:exreg 7
registration
  cluster non empty strict TopGrStr;
end;

:: TOPGRP_1:exreg 8
registration
  cluster non empty trivial TopSpace-like strict TopGrStr;
end;

:: TOPGRP_1:funcnot 6 => TOPGRP_1:func 6
definition
  let a1 be non empty Group-like associative TopGrStr;
  redefine func inverse_op a1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
end;

:: TOPGRP_1:attrnot 3 => TOPGRP_1:attr 3
definition
  let a1 be non empty Group-like associative TopGrStr;
  attr a1 is UnContinuous means
    inverse_op a1 is continuous(a1, a1);
end;

:: TOPGRP_1:dfs 7
definiens
  let a1 be non empty Group-like associative TopGrStr;
To prove
     a1 is UnContinuous
it is sufficient to prove
  thus inverse_op a1 is continuous(a1, a1);

:: TOPGRP_1:def 7
theorem
for b1 being non empty Group-like associative TopGrStr holds
      b1 is UnContinuous
   iff
      inverse_op b1 is continuous(b1, b1);

:: TOPGRP_1:attrnot 4 => TOPGRP_1:attr 4
definition
  let a1 be TopSpace-like TopGrStr;
  attr a1 is BinContinuous means
    for b1 being Function-like quasi_total Relation of the carrier of [:a1,a1:],the carrier of a1
          st b1 = the multF of a1
       holds b1 is continuous([:a1,a1:], a1);
end;

:: TOPGRP_1:dfs 8
definiens
  let a1 be TopSpace-like TopGrStr;
To prove
     a1 is BinContinuous
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of the carrier of [:a1,a1:],the carrier of a1
          st b1 = the multF of a1
       holds b1 is continuous([:a1,a1:], a1);

:: TOPGRP_1:def 8
theorem
for b1 being TopSpace-like TopGrStr holds
      b1 is BinContinuous
   iff
      for b2 being Function-like quasi_total Relation of the carrier of [:b1,b1:],the carrier of b1
            st b2 = the multF of b1
         holds b2 is continuous([:b1,b1:], b1);

:: TOPGRP_1:exreg 9
registration
  cluster non empty trivial TopSpace-like unital Group-like associative commutative strict UnContinuous BinContinuous TopGrStr;
end;

:: TOPGRP_1:modenot 4
definition
  mode TopGroup is non empty TopSpace-like Group-like associative TopGrStr;
end;

:: TOPGRP_1:modenot 5
definition
  mode TopologicalGroup is non empty TopSpace-like Group-like associative UnContinuous BinContinuous TopGrStr;
end;

:: TOPGRP_1:th 37
theorem
for b1 being non empty TopSpace-like BinContinuous TopGrStr
for b2, b3 being Element of the carrier of b1
for b4 being a_neighborhood of b2 * b3 holds
   ex b5 being open a_neighborhood of b2 st
      ex b6 being open a_neighborhood of b3 st
         b5 * b6 c= b4;

:: TOPGRP_1:th 38
theorem
for b1 being non empty TopSpace-like TopGrStr
      st for b2, b3 being Element of the carrier of b1
        for b4 being a_neighborhood of b2 * b3 holds
           ex b5 being a_neighborhood of b2 st
              ex b6 being a_neighborhood of b3 st
                 b5 * b6 c= b4
   holds b1 is BinContinuous;

:: TOPGRP_1:th 39
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous TopGrStr
for b2 being Element of the carrier of b1
for b3 being a_neighborhood of b2 " holds
   ex b4 being open a_neighborhood of b2 st
      b4 " c= b3;

:: TOPGRP_1:th 40
theorem
for b1 being non empty TopSpace-like Group-like associative TopGrStr
      st for b2 being Element of the carrier of b1
        for b3 being a_neighborhood of b2 " holds
           ex b4 being a_neighborhood of b2 st
              b4 " c= b3
   holds b1 is UnContinuous;

:: TOPGRP_1:th 41
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous BinContinuous TopGrStr
for b2, b3 being Element of the carrier of b1
for b4 being a_neighborhood of b2 * (b3 ") holds
   ex b5 being open a_neighborhood of b2 st
      ex b6 being open a_neighborhood of b3 st
         b5 * (b6 ") c= b4;

:: TOPGRP_1:th 42
theorem
for b1 being non empty TopSpace-like Group-like associative TopGrStr
      st for b2, b3 being Element of the carrier of b1
        for b4 being a_neighborhood of b2 * (b3 ") holds
           ex b5 being a_neighborhood of b2 st
              ex b6 being a_neighborhood of b3 st
                 b5 * (b6 ") c= b4
   holds b1 is non empty TopSpace-like Group-like associative UnContinuous BinContinuous TopGrStr;

:: TOPGRP_1:funcreg 12
registration
  let a1 be non empty TopSpace-like BinContinuous TopGrStr;
  let a2 be Element of the carrier of a1;
  cluster a2 * -> Function-like quasi_total continuous;
end;

:: TOPGRP_1:funcreg 13
registration
  let a1 be non empty TopSpace-like BinContinuous TopGrStr;
  let a2 be Element of the carrier of a1;
  cluster * a2 -> Function-like quasi_total continuous;
end;

:: TOPGRP_1:th 43
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being Element of the carrier of b1 holds
   b2 * is Homeomorphism of b1;

:: TOPGRP_1:th 44
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being Element of the carrier of b1 holds
   * b2 is Homeomorphism of b1;

:: TOPGRP_1:funcnot 7 => TOPGRP_1:func 7
definition
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be Element of the carrier of a1;
  redefine func a2 * -> Homeomorphism of a1;
end;

:: TOPGRP_1:funcnot 8 => TOPGRP_1:func 8
definition
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be Element of the carrier of a1;
  redefine func * a2 -> Homeomorphism of a1;
end;

:: TOPGRP_1:th 45
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous TopGrStr holds
   inverse_op b1 is Homeomorphism of b1;

:: TOPGRP_1:funcnot 9 => TOPGRP_1:func 9
definition
  let a1 be non empty TopSpace-like Group-like associative UnContinuous TopGrStr;
  redefine func inverse_op a1 -> Homeomorphism of a1;
end;

:: TOPGRP_1:condreg 7
registration
  cluster non empty TopSpace-like Group-like associative BinContinuous -> homogeneous (TopGrStr);
end;

:: TOPGRP_1:th 46
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being closed Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b2 * b3 is closed(b1);

:: TOPGRP_1:th 47
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being closed Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 * b2 is closed(b1);

:: TOPGRP_1:funcreg 14
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be closed Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a2 * a3 -> closed;
end;

:: TOPGRP_1:funcreg 15
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be closed Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a3 * a2 -> closed;
end;

:: TOPGRP_1:th 48
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous TopGrStr
for b2 being closed Element of bool the carrier of b1 holds
   b2 " is closed(b1);

:: TOPGRP_1:funcreg 16
registration
  let a1 be non empty TopSpace-like Group-like associative UnContinuous TopGrStr;
  let a2 be closed Element of bool the carrier of a1;
  cluster a2 " -> closed;
end;

:: TOPGRP_1:th 49
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being open Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b2 * b3 is open(b1);

:: TOPGRP_1:th 50
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being open Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 * b2 is open(b1);

:: TOPGRP_1:funcreg 17
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be open Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a2 * a3 -> open;
end;

:: TOPGRP_1:funcreg 18
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be open Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a3 * a2 -> open;
end;

:: TOPGRP_1:th 51
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous TopGrStr
for b2 being open Element of bool the carrier of b1 holds
   b2 " is open(b1);

:: TOPGRP_1:funcreg 19
registration
  let a1 be non empty TopSpace-like Group-like associative UnContinuous TopGrStr;
  let a2 be open Element of bool the carrier of a1;
  cluster a2 " -> open;
end;

:: TOPGRP_1:th 52
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2, b3 being Element of bool the carrier of b1
      st b3 is open(b1)
   holds b3 * b2 is open(b1);

:: TOPGRP_1:th 53
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2, b3 being Element of bool the carrier of b1
      st b3 is open(b1)
   holds b2 * b3 is open(b1);

:: TOPGRP_1:funcreg 20
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be open Element of bool the carrier of a1;
  let a3 be Element of bool the carrier of a1;
  cluster a2 * a3 -> open;
end;

:: TOPGRP_1:funcreg 21
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be open Element of bool the carrier of a1;
  let a3 be Element of bool the carrier of a1;
  cluster a3 * a2 -> open;
end;

:: TOPGRP_1:th 54
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous TopGrStr
for b2 being Element of the carrier of b1
for b3 being a_neighborhood of b2 holds
   b3 " is a_neighborhood of b2 ";

:: TOPGRP_1:th 55
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous BinContinuous TopGrStr
for b2 being Element of the carrier of b1
for b3 being a_neighborhood of b2 * (b2 ") holds
   ex b4 being open a_neighborhood of b2 st
      b4 * (b4 ") c= b3;

:: TOPGRP_1:th 56
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous TopGrStr
for b2 being dense Element of bool the carrier of b1 holds
   b2 " is dense(b1);

:: TOPGRP_1:funcreg 22
registration
  let a1 be non empty TopSpace-like Group-like associative UnContinuous TopGrStr;
  let a2 be dense Element of bool the carrier of a1;
  cluster a2 " -> dense;
end;

:: TOPGRP_1:th 57
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being dense Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 * b2 is dense(b1);

:: TOPGRP_1:th 58
theorem
for b1 being non empty TopSpace-like Group-like associative BinContinuous TopGrStr
for b2 being dense Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b2 * b3 is dense(b1);

:: TOPGRP_1:funcreg 23
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be dense Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a2 * a3 -> dense;
end;

:: TOPGRP_1:funcreg 24
registration
  let a1 be non empty TopSpace-like Group-like associative BinContinuous TopGrStr;
  let a2 be dense Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a3 * a2 -> dense;
end;

:: TOPGRP_1:th 59
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous BinContinuous TopGrStr
for b2 being Basis of 1_ b1
for b3 being dense Element of bool the carrier of b1 holds
   {b4 * b5 where b4 is Element of bool the carrier of b1, b5 is Element of the carrier of b1: b4 in b2 & b5 in b3} is Basis of b1;

:: TOPGRP_1:th 60
theorem
for b1 being non empty TopSpace-like Group-like associative UnContinuous BinContinuous TopGrStr holds
   b1 is being_T3;

:: TOPGRP_1:condreg 8
registration
  cluster non empty TopSpace-like Group-like associative UnContinuous BinContinuous -> being_T3 (TopGrStr);
end;

:: TOPGRP_1:th 61
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b1 is empty
   holds b2 is being_homeomorphism(b1, b1);