Article WAYBEL20, MML version 4.99.1005

:: WAYBEL20:th 1
theorem
for b1 being set
for b2 being Element of bool id b1 holds
   proj1 b2 = proj2 b2;

:: WAYBEL20:th 2
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2 holds
   [:b3,b3:] " id b2 is total symmetric transitive Relation of b1,b1;

:: WAYBEL20:funcnot 1 => WAYBEL20:func 1
definition
  let a1, a2, a3, a4 be RelStr;
  let a5 be Function-like quasi_total Relation of the carrier of a1,the carrier of a3;
  let a6 be Function-like quasi_total Relation of the carrier of a2,the carrier of a4;
  redefine func [:a5, a6:] -> Function-like quasi_total Relation of the carrier of [:a1,a2:],the carrier of [:a3,a4:];
end;

:: WAYBEL20:th 3
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set holds
   proj1 ([:b1,b2:] .: b3) c= b1 .: proj1 b3 &
    proj2 ([:b1,b2:] .: b3) c= b2 .: proj2 b3;

:: WAYBEL20:th 4
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set
      st b3 c= [:proj1 b1,proj1 b2:]
   holds proj1 ([:b1,b2:] .: b3) = b1 .: proj1 b3 &
    proj2 ([:b1,b2:] .: b3) = b2 .: proj2 b3;

:: WAYBEL20:th 5
theorem
for b1 being non empty antisymmetric RelStr
      st ex_inf_of {},b1
   holds b1 is upper-bounded;

:: WAYBEL20:th 6
theorem
for b1 being non empty antisymmetric RelStr
      st ex_sup_of {},b1
   holds b1 is lower-bounded;

:: WAYBEL20:th 7
theorem
for b1, b2 being non empty antisymmetric RelStr
for b3 being Element of bool the carrier of [:b1,b2:]
      st ex_inf_of b3,[:b1,b2:]
   holds "/\"(b3,[:b1,b2:]) = ["/\"(proj1 b3,b1),"/\"(proj2 b3,b2)];

:: WAYBEL20:th 8
theorem
for b1, b2 being non empty antisymmetric RelStr
for b3 being Element of bool the carrier of [:b1,b2:]
      st ex_sup_of b3,[:b1,b2:]
   holds "\/"(b3,[:b1,b2:]) = ["\/"(proj1 b3,b1),"\/"(proj2 b3,b2)];

:: WAYBEL20:th 9
theorem
for b1, b2, b3, b4 being non empty antisymmetric RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
      st b5 is infs-preserving(b1, b3) & b6 is infs-preserving(b2, b4)
   holds [:b5,b6:] is infs-preserving([:b1,b2:], [:b3,b4:]);

:: WAYBEL20:th 10
theorem
for b1, b2, b3, b4 being non empty reflexive antisymmetric RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
      st b5 is filtered-infs-preserving(b1, b3) & b6 is filtered-infs-preserving(b2, b4)
   holds [:b5,b6:] is filtered-infs-preserving([:b1,b2:], [:b3,b4:]);

:: WAYBEL20:th 11
theorem
for b1, b2, b3, b4 being non empty antisymmetric RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
      st b5 is sups-preserving(b1, b3) & b6 is sups-preserving(b2, b4)
   holds [:b5,b6:] is sups-preserving([:b1,b2:], [:b3,b4:]);

:: WAYBEL20:th 12
theorem
for b1, b2, b3, b4 being non empty reflexive antisymmetric RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
      st b5 is directed-sups-preserving(b1, b3) & b6 is directed-sups-preserving(b2, b4)
   holds [:b5,b6:] is directed-sups-preserving([:b1,b2:], [:b3,b4:]);

:: WAYBEL20:th 13
theorem
for b1 being non empty antisymmetric RelStr
for b2 being Element of bool the carrier of [:b1,b1:]
      st b2 c= id the carrier of b1 & ex_inf_of b2,[:b1,b1:]
   holds "/\"(b2,[:b1,b1:]) in id the carrier of b1;

:: WAYBEL20:th 14
theorem
for b1 being non empty antisymmetric RelStr
for b2 being Element of bool the carrier of [:b1,b1:]
      st b2 c= id the carrier of b1 & ex_sup_of b2,[:b1,b1:]
   holds "\/"(b2,[:b1,b1:]) in id the carrier of b1;

:: WAYBEL20:th 15
theorem
for b1, b2 being non empty RelStr
      st b1,b2 are_isomorphic & b1 is reflexive
   holds b2 is reflexive;

:: WAYBEL20:th 16
theorem
for b1, b2 being non empty RelStr
      st b1,b2 are_isomorphic & b1 is transitive
   holds b2 is transitive;

:: WAYBEL20:th 17
theorem
for b1, b2 being non empty RelStr
      st b1,b2 are_isomorphic & b1 is antisymmetric
   holds b2 is antisymmetric;

:: WAYBEL20:th 18
theorem
for b1, b2 being non empty RelStr
      st b1,b2 are_isomorphic & b1 is complete
   holds b2 is complete;

:: WAYBEL20:th 19
theorem
for b1 being non empty transitive RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is infs-preserving(b1, b1)
   holds corestr b2 is infs-preserving(b1, Image b2);

:: WAYBEL20:th 20
theorem
for b1 being non empty transitive RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is filtered-infs-preserving(b1, b1)
   holds corestr b2 is filtered-infs-preserving(b1, Image b2);

:: WAYBEL20:th 21
theorem
for b1 being non empty transitive RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is sups-preserving(b1, b1)
   holds corestr b2 is sups-preserving(b1, Image b2);

:: WAYBEL20:th 22
theorem
for b1 being non empty transitive RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is directed-sups-preserving(b1, b1)
   holds corestr b2 is directed-sups-preserving(b1, Image b2);

:: WAYBEL20:th 24
theorem
for b1, b2 being non empty reflexive antisymmetric RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is filtered-infs-preserving(b1, b2)
   holds b3 is monotone(b1, b2);

:: WAYBEL20:th 25
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is monotone(b1, b2)
for b4 being Element of bool the carrier of b1
      st b4 is filtered(b1)
   holds b3 .: b4 is filtered(b2);

:: WAYBEL20:th 26
theorem
for b1, b2, b3 being non empty RelStr
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 b3
      st b4 is infs-preserving(b1, b2) & b5 is infs-preserving(b2, b3)
   holds b5 * b4 is infs-preserving(b1, b3);

:: WAYBEL20:th 27
theorem
for b1, b2, b3 being non empty reflexive antisymmetric RelStr
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 b3
      st b4 is filtered-infs-preserving(b1, b2) & b5 is filtered-infs-preserving(b2, b3)
   holds b5 * b4 is filtered-infs-preserving(b1, b3);

:: WAYBEL20:th 28
theorem
for b1, b2, b3 being non empty RelStr
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 b3
      st b4 is sups-preserving(b1, b2) & b5 is sups-preserving(b2, b3)
   holds b5 * b4 is sups-preserving(b1, b3);

:: WAYBEL20:th 29
theorem
for b1, b2, b3 being non empty reflexive antisymmetric RelStr
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 b3
      st b4 is directed-sups-preserving(b1, b2) & b5 is directed-sups-preserving(b2, b3)
   holds b5 * b4 is directed-sups-preserving(b1, b3);

:: WAYBEL20:th 30
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is antisymmetric lower-bounded RelStr
   holds product b2 is lower-bounded;

:: WAYBEL20:th 31
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is antisymmetric upper-bounded RelStr
   holds product b2 is upper-bounded;

:: WAYBEL20:th 32
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is antisymmetric lower-bounded RelStr
for b3 being Element of b1 holds
   (Bottom product b2) . b3 = Bottom (b2 . b3);

:: WAYBEL20:th 33
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is antisymmetric upper-bounded RelStr
for b3 being Element of b1 holds
   (Top product b2) . b3 = Top (b2 . b3);

:: WAYBEL20:th 34
theorem
for b1 being non empty set
for b2 being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
   holds product b2 is continuous;

:: WAYBEL20:th 35
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b3 being CLHomomorphism of b1,b2
for b4 being Element of bool the carrier of [:b1,b1:]
      st b4 = [:b3,b3:] " id the carrier of b2
   holds subrelstr b4 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of [:b1,b1:];

:: WAYBEL20:funcnot 2 => WAYBEL20:func 2
definition
  let a1 be RelStr;
  let a2 be Element of bool the carrier of [:a1,a1:];
  assume a2 is total symmetric transitive Relation of the carrier of a1,the carrier of a1;
  func EqRel A2 -> total symmetric transitive Relation of the carrier of a1,the carrier of a1 equals
    a2;
end;

:: WAYBEL20:def 1
theorem
for b1 being RelStr
for b2 being Element of bool the carrier of [:b1,b1:]
      st b2 is total symmetric transitive Relation of the carrier of b1,the carrier of b1
   holds EqRel b2 = b2;

:: WAYBEL20:attrnot 1 => WAYBEL20:attr 1
definition
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of [:a1,a1:];
  attr a2 is CLCongruence means
    a2 is total symmetric transitive Relation of the carrier of a1,the carrier of a1 &
     subrelstr a2 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of [:a1,a1:];
end;

:: WAYBEL20:dfs 2
definiens
  let a1 be non empty RelStr;
  let a2 be Element of bool the carrier of [:a1,a1:];
To prove
     a2 is CLCongruence
it is sufficient to prove
  thus a2 is total symmetric transitive Relation of the carrier of a1,the carrier of a1 &
     subrelstr a2 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of [:a1,a1:];

:: WAYBEL20:def 2
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of [:b1,b1:] holds
      b2 is CLCongruence(b1)
   iff
      b2 is total symmetric transitive Relation of the carrier of b1,the carrier of b1 &
       subrelstr b2 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of [:b1,b1:];

:: WAYBEL20:th 36
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty Element of bool the carrier of [:b1,b1:]
   st b2 is CLCongruence(b1)
for b3 being Element of the carrier of b1 holds
   ["/\"(Class(EqRel b2,b3),b1),b3] in b2;

:: WAYBEL20:funcnot 3 => WAYBEL20:func 3
definition
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  let a2 be non empty Element of bool the carrier of [:a1,a1:];
  assume a2 is CLCongruence(a1);
  func kernel_op A2 -> Function-like quasi_total kernel Relation of the carrier of a1,the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = "/\"(Class(EqRel a2,b1),a1);
end;

:: WAYBEL20:def 3
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty Element of bool the carrier of [:b1,b1:]
   st b2 is CLCongruence(b1)
for b3 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1 holds
      b3 = kernel_op b2
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = "/\"(Class(EqRel b2,b4),b1);

:: WAYBEL20:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty Element of bool the carrier of [:b1,b1:]
      st b2 is CLCongruence(b1)
   holds kernel_op b2 is directed-sups-preserving(b1, b1) &
    b2 = [:kernel_op b2,kernel_op b2:] " id the carrier of b1;

:: WAYBEL20:th 38
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Element of bool the carrier of [:b1,b1:]
for b3 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1
      st b3 is directed-sups-preserving(b1, b1) &
         b2 = [:b3,b3:] " id the carrier of b1
   holds ex b4 being strict reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr st
      the carrier of b4 = Class EqRel b2 &
       the InternalRel of b4 = {[Class(EqRel b2,b5),Class(EqRel b2,b6)] where b5 is Element of the carrier of b1, b6 is Element of the carrier of b1: b3 . b5 <= b3 . b6} &
       (for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b4
             st for b6 being Element of the carrier of b1 holds
                  b5 . b6 = Class(EqRel b2,b6)
          holds b5 is CLHomomorphism of b1,b4);

:: WAYBEL20:th 39
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Element of bool the carrier of [:b1,b1:]
      st b2 is total symmetric transitive Relation of the carrier of b1,the carrier of b1 &
         (ex b3 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr st
            the carrier of b3 = Class EqRel b2 &
             (for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
                   st for b5 being Element of the carrier of b1 holds
                        b4 . b5 = Class(EqRel b2,b5)
                holds b4 is CLHomomorphism of b1,b3))
   holds subrelstr b2 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of [:b1,b1:];

:: WAYBEL20:exreg 1
registration
  let a1 be non empty reflexive RelStr;
  cluster non empty Relation-like Function-like quasi_total total directed-sups-preserving kernel Relation of the carrier of a1,the carrier of a1;
end;

:: WAYBEL20:funcnot 4 => WAYBEL20:func 4
definition
  let a1 be non empty reflexive RelStr;
  let a2 be Function-like quasi_total kernel Relation of the carrier of a1,the carrier of a1;
  func kernel_congruence A2 -> non empty Element of bool the carrier of [:a1,a1:] equals
    [:a2,a2:] " id the carrier of a1;
end;

:: WAYBEL20:def 4
theorem
for b1 being non empty reflexive RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1 holds
   kernel_congruence b2 = [:b2,b2:] " id the carrier of b1;

:: WAYBEL20:th 40
theorem
for b1 being non empty reflexive RelStr
for b2 being Function-like quasi_total kernel Relation of the carrier of b1,the carrier of b1 holds
   kernel_congruence b2 is total symmetric transitive Relation of the carrier of b1,the carrier of b1;

:: WAYBEL20:th 41
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Function-like quasi_total directed-sups-preserving kernel Relation of the carrier of b1,the carrier of b1 holds
   kernel_congruence b2 is CLCongruence(b1);

:: WAYBEL20:funcnot 5 => WAYBEL20:func 5
definition
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr;
  let a2 be non empty Element of bool the carrier of [:a1,a1:];
  assume a2 is CLCongruence(a1);
  func A1 ./. A2 -> strict reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr means
    the carrier of it = Class EqRel a2 &
     (for b1, b2 being Element of the carrier of it holds
        b1 <= b2
     iff
        "/\"(b1,a1) <= "/\"(b2,a1));
end;

:: WAYBEL20:def 5
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being non empty Element of bool the carrier of [:b1,b1:]
   st b2 is CLCongruence(b1)
for b3 being strict reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr holds
      b3 = b1 ./. b2
   iff
      the carrier of b3 = Class EqRel b2 &
       (for b4, b5 being Element of the carrier of b3 holds
          b4 <= b5
       iff
          "/\"(b4,b1) <= "/\"(b5,b1));

:: WAYBEL20:th 42
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being non empty Element of bool the carrier of [:b1,b1:]
   st b2 is CLCongruence(b1)
for b3 being set holds
      b3 is Element of the carrier of b1 ./. b2
   iff
      ex b4 being Element of the carrier of b1 st
         b3 = Class(EqRel b2,b4);

:: WAYBEL20:th 43
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being non empty Element of bool the carrier of [:b1,b1:]
      st b2 is CLCongruence(b1)
   holds b2 = kernel_congruence kernel_op b2;

:: WAYBEL20:th 44
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Function-like quasi_total directed-sups-preserving kernel Relation of the carrier of b1,the carrier of b1 holds
   b2 = kernel_op kernel_congruence b2;

:: WAYBEL20:th 45
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
for b2 being Function-like quasi_total projection Relation of the carrier of b1,the carrier of b1
      st b2 is infs-preserving(b1, b1)
   holds Image b2 is reflexive transitive antisymmetric with_suprema with_infima continuous RelStr &
    Image b2 is infs-inheriting(b1);