Article YELLOW16, MML version 4.99.1005

:: YELLOW16:th 1
theorem
for b1, b2 being Relation-like set holds
b1 * b2 = b1 (#) b2;

:: YELLOW16:th 2
theorem
for b1 being set
for b2 being non empty RelStr
for b3 being non empty SubRelStr of b2
for b4, b5 being Function-like quasi_total Relation of b1,the carrier of b3
for b6, b7 being Function-like quasi_total Relation of b1,the carrier of b2
      st b6 = b4 & b7 = b5 & b4 <= b5
   holds b6 <= b7;

:: YELLOW16:th 3
theorem
for b1 being set
for b2 being non empty RelStr
for b3 being non empty full SubRelStr of b2
for b4, b5 being Function-like quasi_total Relation of b1,the carrier of b3
for b6, b7 being Function-like quasi_total Relation of b1,the carrier of b2
      st b6 = b4 & b7 = b5 & b6 <= b7
   holds b4 <= b5;

:: YELLOW16:exreg 1
registration
  let a1 be non empty RelStr;
  let a2 be non empty reflexive antisymmetric RelStr;
  cluster Relation-like Function-like non empty total quasi_total monotone directed-sups-preserving Relation of the carrier of a1,the carrier of a2;
end;

:: YELLOW16:th 4
theorem
for b1, b2 being Relation-like Function-like set
      st b1 is idempotent & proj2 b2 c= proj2 b1 & proj2 b2 c= proj1 b1
   holds b2 * b1 = b2;

:: YELLOW16:exreg 2
registration
  let a1 be 1-sorted;
  cluster Relation-like Function-like quasi_total idempotent Relation of the carrier of a1,the carrier of a1;
end;

:: YELLOW16:th 5
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty full directed-sups-inheriting SubRelStr of b1 holds
   b2 is up-complete;

:: YELLOW16:th 6
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is idempotent & b2 is directed-sups-preserving(b1, b1)
   holds Image b2 is directed-sups-inheriting(b1);

:: YELLOW16:th 9
theorem
for b1 being non empty RelStr
for b2 being non empty SubRelStr of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 * incl(b2,b1) = id b2
   holds b3 is Function-like quasi_total idempotent Relation of the carrier of b1,the carrier of b1;

:: YELLOW16:prednot 1 => YELLOW16:pred 1
definition
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
  let a3 be Relation-like Function-like set;
  pred A3 is_a_retraction_of A2,A1 means
    a3 is Function-like quasi_total directed-sups-preserving Relation of the carrier of a2,the carrier of a1 &
     a3 | the carrier of a1 = id a1 &
     a1 is full directed-sups-inheriting SubRelStr of a2;
end;

:: YELLOW16:dfs 1
definiens
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
  let a3 be Relation-like Function-like set;
To prove
     a3 is_a_retraction_of a2,a1
it is sufficient to prove
  thus a3 is Function-like quasi_total directed-sups-preserving Relation of the carrier of a2,the carrier of a1 &
     a3 | the carrier of a1 = id a1 &
     a1 is full directed-sups-inheriting SubRelStr of a2;

:: YELLOW16:def 1
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Relation-like Function-like set holds
      b3 is_a_retraction_of b2,b1
   iff
      b3 is Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b1 &
       b3 | the carrier of b1 = id b1 &
       b1 is full directed-sups-inheriting SubRelStr of b2;

:: YELLOW16:prednot 2 => YELLOW16:pred 2
definition
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
  let a3 be Relation-like Function-like set;
  pred A3 is_an_UPS_retraction_of A2,A1 means
    a3 is Function-like quasi_total directed-sups-preserving Relation of the carrier of a2,the carrier of a1 &
     (ex b1 being Function-like quasi_total directed-sups-preserving Relation of the carrier of a1,the carrier of a2 st
        b1 * a3 = id a1);
end;

:: YELLOW16:dfs 2
definiens
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
  let a3 be Relation-like Function-like set;
To prove
     a3 is_an_UPS_retraction_of a2,a1
it is sufficient to prove
  thus a3 is Function-like quasi_total directed-sups-preserving Relation of the carrier of a2,the carrier of a1 &
     (ex b1 being Function-like quasi_total directed-sups-preserving Relation of the carrier of a1,the carrier of a2 st
        b1 * a3 = id a1);

:: YELLOW16:def 2
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Relation-like Function-like set holds
      b3 is_an_UPS_retraction_of b2,b1
   iff
      b3 is Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b1 &
       (ex b4 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b1,the carrier of b2 st
          b4 * b3 = id b1);

:: YELLOW16:prednot 3 => YELLOW16:pred 3
definition
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
  pred A1 is_a_retract_of A2 means
    ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a1 st
       b1 is_a_retraction_of a2,a1;
end;

:: YELLOW16:dfs 3
definiens
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
To prove
     a1 is_a_retract_of a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a1 st
       b1 is_a_retraction_of a2,a1;

:: YELLOW16:def 3
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr holds
   b1 is_a_retract_of b2
iff
   ex b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
      b3 is_a_retraction_of b2,b1;

:: YELLOW16:prednot 4 => YELLOW16:pred 4
definition
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
  pred A1 is_an_UPS_retract_of A2 means
    ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a1 st
       b1 is_an_UPS_retraction_of a2,a1;
end;

:: YELLOW16:dfs 4
definiens
  let a1, a2 be non empty reflexive transitive antisymmetric RelStr;
To prove
     a1 is_an_UPS_retract_of a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a1 st
       b1 is_an_UPS_retraction_of a2,a1;

:: YELLOW16:def 4
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr holds
   b1 is_an_UPS_retract_of b2
iff
   ex b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
      b3 is_an_UPS_retraction_of b2,b1;

:: YELLOW16:th 10
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Relation-like Function-like set
      st b3 is_a_retraction_of b2,b1
   holds (incl(b1,b2)) * b3 = id b1;

:: YELLOW16:th 11
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Relation-like Function-like set
      st b3 is_a_retraction_of b2,b1
   holds b3 is_an_UPS_retraction_of b2,b1;

:: YELLOW16:th 12
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Relation-like Function-like set
      st b3 is_a_retraction_of b2,b1
   holds proj2 b3 = the carrier of b1;

:: YELLOW16:th 13
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Relation-like Function-like set
      st b3 is_an_UPS_retraction_of b2,b1
   holds proj2 b3 = the carrier of b1;

:: YELLOW16:th 14
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Relation-like Function-like set
      st b3 is_a_retraction_of b2,b1
   holds b3 is Function-like quasi_total idempotent Relation of the carrier of b2,the carrier of b2;

:: YELLOW16:th 15
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b3 is_a_retraction_of b1,b2
   holds Image b3 = RelStr(#the carrier of b2,the InternalRel of b2#);

:: YELLOW16:th 16
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b3 is_a_retraction_of b1,b2
   holds b3 is directed-sups-preserving(b1, b1) & b3 is projection(b1);

:: YELLOW16:th 17
theorem
for b1, b2 being non empty reflexive transitive RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is isomorphic(b1, b2)
   iff
      b3 is monotone(b1, b2) &
       (ex b4 being Function-like quasi_total monotone Relation of the carrier of b2,the carrier of b1 st
          b3 * b4 = id b2 & b4 * b3 = id b1);

:: YELLOW16:th 18
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr holds
   b1,b2 are_isomorphic
iff
   ex b3 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2 st
      ex b4 being Function-like quasi_total monotone Relation of the carrier of b2,the carrier of b1 st
         b3 * b4 = id b2 & b4 * b3 = id b1;

:: YELLOW16:th 19
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
      st b1,b2 are_isomorphic
   holds b1 is_an_UPS_retract_of b2 & b2 is_an_UPS_retract_of b1;

:: YELLOW16:th 20
theorem
for b1, b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total monotone Relation of the carrier of b2,the carrier of b1
      st b3 * b4 = id b2
   holds ex b5 being Function-like quasi_total projection Relation of the carrier of b1,the carrier of b1 st
      b5 = b4 * b3 &
       b5 | the carrier of Image b5 = id Image b5 &
       b2,Image b5 are_isomorphic;

:: YELLOW16:th 21
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Relation-like Function-like set
      st b3 is_an_UPS_retraction_of b1,b2
   holds ex b4 being Function-like quasi_total directed-sups-preserving projection Relation of the carrier of b1,the carrier of b1 st
      b4 is_a_retraction_of b1,Image b4 & b2,Image b4 are_isomorphic;

:: YELLOW16:th 22
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
      st b2 is_a_retract_of b1
   holds b2 is up-complete;

:: YELLOW16:th 23
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
      st b2 is_a_retract_of b1
   holds b2 is complete;

:: YELLOW16:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
      st b2 is_a_retract_of b1
   holds b2 is continuous;

:: YELLOW16:th 25
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
      st b2 is_an_UPS_retract_of b1
   holds b2 is up-complete;

:: YELLOW16:th 26
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
      st b2 is_an_UPS_retract_of b1
   holds b2 is complete;

:: YELLOW16:th 27
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
      st b2 is_an_UPS_retract_of b1
   holds b2 is continuous;

:: YELLOW16:th 28
theorem
for b1 being RelStr
for b2 being full SubRelStr of b1
for b3 being SubRelStr of b2 holds
      b3 is full(b2)
   iff
      b3 is full SubRelStr of b1;

:: YELLOW16:th 29
theorem
for b1 being non empty transitive RelStr
for b2 being non empty full directed-sups-inheriting SubRelStr of b1
for b3 being non empty directed-sups-inheriting SubRelStr of b2 holds
   b3 is directed-sups-inheriting SubRelStr of b1;

:: YELLOW16:th 30
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2, b3 being non empty full directed-sups-inheriting SubRelStr of b1
      st b2 is SubRelStr of b3
   holds b2 is full directed-sups-inheriting SubRelStr of b3;

:: YELLOW16:attrnot 1 => YELLOW16:attr 1
definition
  let a1 be Relation-like set;
  attr a1 is Poset-yielding means
    for b1 being set
          st b1 in proj2 a1
       holds b1 is reflexive transitive antisymmetric RelStr;
end;

:: YELLOW16:dfs 5
definiens
  let a1 be Relation-like set;
To prove
     a1 is Poset-yielding
it is sufficient to prove
  thus for b1 being set
          st b1 in proj2 a1
       holds b1 is reflexive transitive antisymmetric RelStr;

:: YELLOW16:def 5
theorem
for b1 being Relation-like set holds
      b1 is Poset-yielding
   iff
      for b2 being set
            st b2 in proj2 b1
         holds b2 is reflexive transitive antisymmetric RelStr;

:: YELLOW16:condreg 1
registration
  cluster Relation-like Poset-yielding -> RelStr-yielding reflexive-yielding (set);
end;

:: YELLOW16:funcnot 1 => YELLOW16:func 1
definition
  let a1 be non empty set;
  let a2 be non empty RelStr;
  let a3 be Element of a1;
  let a4 be Element of bool the carrier of a2 |^ a1;
  redefine func pi(a4,a3) -> Element of bool the carrier of a2;
end;

:: YELLOW16:funcreg 1
registration
  let a1 be set;
  let a2 be reflexive transitive antisymmetric RelStr;
  cluster a1 --> a2 -> Poset-yielding;
end;

:: YELLOW16:exreg 3
registration
  let a1 be set;
  cluster Relation-like Function-like non-Empty Poset-yielding ManySortedSet of a1;
end;

:: YELLOW16:funcreg 2
registration
  let a1 be non empty set;
  let a2 be non-Empty Poset-yielding ManySortedSet of a1;
  cluster product a2 -> strict transitive antisymmetric;
end;

:: YELLOW16:funcnot 2 => YELLOW16:func 2
definition
  let a1 be non empty set;
  let a2 be non-Empty Poset-yielding ManySortedSet of a1;
  let a3 be Element of a1;
  redefine func a2 . a3 -> non empty reflexive transitive antisymmetric RelStr;
end;

:: YELLOW16:th 31
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
for b3 being Element of the carrier of product b2
for b4 being Element of bool the carrier of product b2 holds
      b4 is_<=_than b3
   iff
      for b5 being Element of b1 holds
         pi(b4,b5) is_<=_than b3 . b5;

:: YELLOW16:th 32
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
for b3 being Element of the carrier of product b2
for b4 being Element of bool the carrier of product b2 holds
      b3 is_<=_than b4
   iff
      for b5 being Element of b1 holds
         b3 . b5 is_<=_than pi(b4,b5);

:: YELLOW16:th 33
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
for b3 being Element of bool the carrier of product b2 holds
      ex_sup_of b3,product b2
   iff
      for b4 being Element of b1 holds
         ex_sup_of pi(b3,b4),b2 . b4;

:: YELLOW16:th 34
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
for b3 being Element of bool the carrier of product b2 holds
      ex_inf_of b3,product b2
   iff
      for b4 being Element of b1 holds
         ex_inf_of pi(b3,b4),b2 . b4;

:: YELLOW16:th 35
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
for b3 being Element of bool the carrier of product b2
   st ex_sup_of b3,product b2
for b4 being Element of b1 holds
   ("\/"(b3,product b2)) . b4 = "\/"(pi(b3,b4),b2 . b4);

:: YELLOW16:th 36
theorem
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1
for b3 being Element of bool the carrier of product b2
   st ex_inf_of b3,product b2
for b4 being Element of b1 holds
   ("/\"(b3,product b2)) . b4 = "/\"(pi(b3,b4),b2 . b4);

:: YELLOW16:th 37
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of b1
for b3 being directed Element of bool the carrier of product b2
for b4 being Element of b1 holds
   pi(b3,b4) is directed(b2 . b4);

:: YELLOW16:th 38
theorem
for b1 being non empty set
for b2, b3 being RelStr-yielding non-Empty ManySortedSet of b1
      st for b4 being Element of b1 holds
           b3 . b4 is SubRelStr of b2 . b4
   holds product b3 is SubRelStr of product b2;

:: YELLOW16:th 39
theorem
for b1 being non empty set
for b2, b3 being RelStr-yielding non-Empty ManySortedSet of b1
      st for b4 being Element of b1 holds
           b3 . b4 is full SubRelStr of b2 . b4
   holds product b3 is full SubRelStr of product b2;

:: YELLOW16:th 40
theorem
for b1 being non empty RelStr
for b2 being non empty SubRelStr of b1
for b3 being set holds
   b2 |^ b3 is SubRelStr of b1 |^ b3;

:: YELLOW16:th 41
theorem
for b1 being non empty RelStr
for b2 being non empty full SubRelStr of b1
for b3 being set holds
   b2 |^ b3 is full SubRelStr of b1 |^ b3;

:: YELLOW16:prednot 5 => YELLOW16:pred 5
definition
  let a1, a2 be non empty RelStr;
  let a3 be set;
  pred A1 inherits_sup_of A3,A2 means
    (ex_sup_of a3,a2) implies "\/"(a3,a2) in the carrier of a1;
end;

:: YELLOW16:dfs 6
definiens
  let a1, a2 be non empty RelStr;
  let a3 be set;
To prove
     a1 inherits_sup_of a3,a2
it is sufficient to prove
  thus (ex_sup_of a3,a2) implies "\/"(a3,a2) in the carrier of a1;

:: YELLOW16:def 6
theorem
for b1, b2 being non empty RelStr
for b3 being set holds
      b1 inherits_sup_of b3,b2
   iff
      (ex_sup_of b3,b2 implies "\/"(b3,b2) in the carrier of b1);

:: YELLOW16:prednot 6 => YELLOW16:pred 6
definition
  let a1, a2 be non empty RelStr;
  let a3 be set;
  pred A1 inherits_inf_of A3,A2 means
    (ex_inf_of a3,a2) implies "/\"(a3,a2) in the carrier of a1;
end;

:: YELLOW16:dfs 7
definiens
  let a1, a2 be non empty RelStr;
  let a3 be set;
To prove
     a1 inherits_inf_of a3,a2
it is sufficient to prove
  thus (ex_inf_of a3,a2) implies "/\"(a3,a2) in the carrier of a1;

:: YELLOW16:def 7
theorem
for b1, b2 being non empty RelStr
for b3 being set holds
      b1 inherits_inf_of b3,b2
   iff
      (ex_inf_of b3,b2 implies "/\"(b3,b2) in the carrier of b1);

:: YELLOW16:th 42
theorem
for b1 being non empty transitive RelStr
for b2 being non empty full SubRelStr of b1
for b3 being Element of bool the carrier of b2 holds
      b2 inherits_sup_of b3,b1
   iff
      (ex_sup_of b3,b1 implies ex_sup_of b3,b2 & "\/"(b3,b2) = "\/"(b3,b1));

:: YELLOW16:th 43
theorem
for b1 being non empty transitive RelStr
for b2 being non empty full SubRelStr of b1
for b3 being Element of bool the carrier of b2 holds
      b2 inherits_inf_of b3,b1
   iff
      (ex_inf_of b3,b1 implies ex_inf_of b3,b2 & "/\"(b3,b2) = "/\"(b3,b1));

:: YELLOW16:sch 1
scheme YELLOW16:sch 1
{F1 -> non empty set,
  F2 -> non-Empty Poset-yielding ManySortedSet of F1(),
  F3 -> non-Empty Poset-yielding ManySortedSet of F1()}:
for b1 being Element of bool the carrier of product F3()
      st P1[b1, product F3()]
   holds product F3() inherits_sup_of b1,product F2()
provided
   for b1 being Element of bool the carrier of product F3()
      st P1[b1, product F3()]
   for b2 being Element of F1() holds
      P1[pi(b1,b2), F3() . b2]
and
   for b1 being Element of F1() holds
      F3() . b1 is full SubRelStr of F2() . b1
and
   for b1 being Element of F1()
   for b2 being Element of bool the carrier of F3() . b1
         st P1[b2, F3() . b1]
      holds F3() . b1 inherits_sup_of b2,F2() . b1;


:: YELLOW16:sch 2
scheme YELLOW16:sch 2
{F1 -> non empty set,
  F2 -> non empty reflexive transitive antisymmetric RelStr,
  F3 -> non empty full SubRelStr of F2()}:
for b1 being Element of bool the carrier of F3() |^ F1()
      st P1[b1, F3() |^ F1()]
   holds F3() |^ F1() inherits_sup_of b1,F2() |^ F1()
provided
   for b1 being Element of bool the carrier of F3() |^ F1()
      st P1[b1, F3() |^ F1()]
   for b2 being Element of F1() holds
      P1[pi(b1,b2), F3()]
and
   for b1 being Element of bool the carrier of F3()
         st P1[b1, F3()]
      holds F3() inherits_sup_of b1,F2();


:: YELLOW16:sch 3
scheme YELLOW16:sch 3
{F1 -> non empty set,
  F2 -> non-Empty Poset-yielding ManySortedSet of F1(),
  F3 -> non-Empty Poset-yielding ManySortedSet of F1()}:
for b1 being Element of bool the carrier of product F3()
      st P1[b1, product F3()]
   holds product F3() inherits_inf_of b1,product F2()
provided
   for b1 being Element of bool the carrier of product F3()
      st P1[b1, product F3()]
   for b2 being Element of F1() holds
      P1[pi(b1,b2), F3() . b2]
and
   for b1 being Element of F1() holds
      F3() . b1 is full SubRelStr of F2() . b1
and
   for b1 being Element of F1()
   for b2 being Element of bool the carrier of F3() . b1
         st P1[b2, F3() . b1]
      holds F3() . b1 inherits_inf_of b2,F2() . b1;


:: YELLOW16:sch 4
scheme YELLOW16:sch 4
{F1 -> non empty set,
  F2 -> non empty reflexive transitive antisymmetric RelStr,
  F3 -> non empty full SubRelStr of F2()}:
for b1 being Element of bool the carrier of F3() |^ F1()
      st P1[b1, F3() |^ F1()]
   holds F3() |^ F1() inherits_inf_of b1,F2() |^ F1()
provided
   for b1 being Element of bool the carrier of F3() |^ F1()
      st P1[b1, F3() |^ F1()]
   for b2 being Element of F1() holds
      P1[pi(b1,b2), F3()]
and
   for b1 being Element of bool the carrier of F3()
         st P1[b1, F3()]
      holds F3() inherits_inf_of b1,F2();


:: YELLOW16:funcreg 3
registration
  let a1 be set;
  let a2 be non empty RelStr;
  let a3 be non empty Element of bool the carrier of a2 |^ a1;
  let a4 be set;
  cluster pi(a3,a4) -> non empty;
end;

:: YELLOW16:th 44
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty full directed-sups-inheriting SubRelStr of b1
for b3 being non empty set holds
   b2 |^ b3 is directed-sups-inheriting SubRelStr of b1 |^ b3;

:: YELLOW16:funcreg 4
registration
  let a1 be non empty set;
  let a2 be RelStr-yielding non-Empty ManySortedSet of a1;
  let a3 be non empty Element of bool the carrier of product a2;
  let a4 be set;
  cluster pi(a3,a4) -> non empty;
end;

:: YELLOW16:th 45
theorem
for b1 being non empty set
for b2 being non empty reflexive transitive antisymmetric up-complete RelStr holds
   b2 |^ b1 is up-complete;

:: YELLOW16:funcreg 5
registration
  let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
  let a2 be non empty set;
  cluster a1 |^ a2 -> strict up-complete;
end;

:: YELLOW16:th 46
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
      st b3 is being_a_retraction(b1, b2)
   holds rng b3 = the carrier of b2;

:: YELLOW16:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for 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)
   holds b3 is idempotent;

:: YELLOW16:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being open Element of bool the carrier of b1 holds
   chi(b2,the carrier of b1) is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of Sierpinski_Space;

:: YELLOW16:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
      st for b4 being open Element of bool the carrier of b1
              st b3 in b4
           holds b2 in b4
   holds (0,1)-->(b3,b2) is Function-like quasi_total continuous Relation of the carrier of Sierpinski_Space,the carrier of b1;

:: YELLOW16:th 50
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being open Element of bool the carrier of b1
      st b2 in b4 & not b3 in b4
   holds (chi(b4,the carrier of b1)) * ((0,1)-->(b3,b2)) = id Sierpinski_Space;

:: YELLOW16:th 51
theorem
for b1 being non empty 1-sorted
for b2, b3 being Element of bool the carrier of b1
for b4 being TopAugmentation of BoolePoset 1
for b5, b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b4
      st b5 = chi(b2,the carrier of b1) & b6 = chi(b3,the carrier of b1)
   holds    b2 c= b3
   iff
      b5 <= b6;

:: YELLOW16:th 52
theorem
for b1 being non empty RelStr
for b2 being non empty set
for b3 being non empty full SubRelStr of b1 |^ b2
      st for b4 being set holds
              b4 is Element of the carrier of b3
           iff
              ex b5 being Element of the carrier of b1 st
                 b4 = b2 --> b5
   holds b1,b3 are_isomorphic;

:: YELLOW16:th 53
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
   b1,b2 are_homeomorphic
iff
   ex b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
      ex b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1 st
         b3 * b4 = id b2 & b4 * b3 = id b1;

:: YELLOW16:th 54
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b4 * b5 = id b2 & b6 is being_homeomorphism(b2, b3)
   holds (b6 * b4) * (b5 * (b6 /")) = id b3;

:: YELLOW16:th 55
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
      st b2 is_Retract_of b1 & b2,b3 are_homeomorphic
   holds b3 is_Retract_of b1;

:: YELLOW16:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
   incl(b2,b1) is continuous(b2, b1);

:: YELLOW16:th 57
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for 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)
   holds b3 * incl(b2,b1) = id b2;

:: YELLOW16:th 58
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
      st b2 is_a_retract_of b1
   holds b2 is_Retract_of b1;

:: YELLOW16:th 59
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
   b1 is_Retract_of b2
iff
   ex b3 being non empty SubSpace of b2 st
      b3 is_a_retract_of b2 & b3,b1 are_homeomorphic;