Article LATSUM_1, MML version 4.99.1005

:: LATSUM_1:th 1
theorem
for b1, b2, b3, b4 being set
      st b1 in b3 \/ b4 & b2 in b3 \/ b4 & (b1 in b3 \ b4 implies not b2 in b3 \ b4) & (b1 in b4 implies not b2 in b4) & (b1 in b3 \ b4 implies not b2 in b4)
   holds b1 in b4 & b2 in b3 \ b4;

:: LATSUM_1:prednot 1 => LATSUM_1:pred 1
definition
  let a1, a2 be RelStr;
  pred A1 tolerates A2 means
    for b1, b2 being set
          st b1 in (the carrier of a1) /\ the carrier of a2 & b2 in (the carrier of a1) /\ the carrier of a2
       holds    [b1,b2] in the InternalRel of a1
       iff
          [b1,b2] in the InternalRel of a2;
end;

:: LATSUM_1:dfs 1
definiens
  let a1, a2 be RelStr;
To prove
     a1 tolerates a2
it is sufficient to prove
  thus for b1, b2 being set
          st b1 in (the carrier of a1) /\ the carrier of a2 & b2 in (the carrier of a1) /\ the carrier of a2
       holds    [b1,b2] in the InternalRel of a1
       iff
          [b1,b2] in the InternalRel of a2;

:: LATSUM_1:def 1
theorem
for b1, b2 being RelStr holds
   b1 tolerates b2
iff
   for b3, b4 being set
         st b3 in (the carrier of b1) /\ the carrier of b2 & b4 in (the carrier of b1) /\ the carrier of b2
      holds    [b3,b4] in the InternalRel of b1
      iff
         [b3,b4] in the InternalRel of b2;

:: LATSUM_1:funcnot 1 => LATSUM_1:func 1
definition
  let a1, a2 be RelStr;
  func A1 [*] A2 -> strict RelStr means
    the carrier of it = (the carrier of a1) \/ the carrier of a2 &
     the InternalRel of it = ((the InternalRel of a1) \/ the InternalRel of a2) \/ ((the InternalRel of a1) * the InternalRel of a2);
end;

:: LATSUM_1:def 2
theorem
for b1, b2 being RelStr
for b3 being strict RelStr holds
      b3 = b1 [*] b2
   iff
      the carrier of b3 = (the carrier of b1) \/ the carrier of b2 &
       the InternalRel of b3 = ((the InternalRel of b1) \/ the InternalRel of b2) \/ ((the InternalRel of b1) * the InternalRel of b2);

:: LATSUM_1:funcreg 1
registration
  let a1 be RelStr;
  let a2 be non empty RelStr;
  cluster a1 [*] a2 -> non empty strict;
end;

:: LATSUM_1:funcreg 2
registration
  let a1 be non empty RelStr;
  let a2 be RelStr;
  cluster a1 [*] a2 -> non empty strict;
end;

:: LATSUM_1:th 2
theorem
for b1, b2 being RelStr holds
the InternalRel of b1 c= the InternalRel of b1 [*] b2 & the InternalRel of b2 c= the InternalRel of b1 [*] b2;

:: LATSUM_1:th 3
theorem
for b1, b2 being RelStr
      st b1 is reflexive & b2 is reflexive
   holds b1 [*] b2 is reflexive;

:: LATSUM_1:th 4
theorem
for b1, b2 being RelStr
for b3, b4 being set
      st [b3,b4] in the InternalRel of b1 [*] b2 & b3 in the carrier of b1 & b4 in the carrier of b1 & b1 tolerates b2 & b1 is transitive
   holds [b3,b4] in the InternalRel of b1;

:: LATSUM_1:th 5
theorem
for b1, b2 being RelStr
for b3, b4 being set
      st [b3,b4] in the InternalRel of b1 [*] b2 & b3 in the carrier of b2 & b4 in the carrier of b2 & b1 tolerates b2 & b2 is transitive
   holds [b3,b4] in the InternalRel of b2;

:: LATSUM_1:th 6
theorem
for b1, b2 being RelStr
for b3, b4 being set holds
([b3,b4] in the InternalRel of b1 implies [b3,b4] in the InternalRel of b1 [*] b2) &
 ([b3,b4] in the InternalRel of b2 implies [b3,b4] in the InternalRel of b1 [*] b2);

:: LATSUM_1:th 7
theorem
for b1, b2 being non empty RelStr
for b3 being Element of the carrier of b1 [*] b2
      st not b3 in the carrier of b1
   holds b3 in (the carrier of b2) \ the carrier of b1;

:: LATSUM_1:th 8
theorem
for b1, b2 being non empty RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b1 [*] b2
      st b3 = b5 & b4 = b6 & b1 tolerates b2 & b1 is transitive
   holds    b3 <= b4
   iff
      b5 <= b6;

:: LATSUM_1:th 9
theorem
for b1, b2 being non empty RelStr
for b3, b4 being Element of the carrier of b1 [*] b2
for b5, b6 being Element of the carrier of b2
      st b3 = b5 & b4 = b6 & b1 tolerates b2 & b2 is transitive
   holds    b3 <= b4
   iff
      b5 <= b6;

:: LATSUM_1:th 10
theorem
for b1, b2 being non empty reflexive transitive antisymmetric with_suprema RelStr
for b3 being set
      st b3 in the carrier of b1
   holds b3 is Element of the carrier of b1 [*] b2;

:: LATSUM_1:th 11
theorem
for b1, b2 being non empty reflexive transitive antisymmetric with_suprema RelStr
for b3 being set
      st b3 in the carrier of b2
   holds b3 is Element of the carrier of b1 [*] b2;

:: LATSUM_1:th 12
theorem
for b1, b2 being non empty RelStr
for b3 being set
      st b3 in (the carrier of b1) /\ the carrier of b2
   holds b3 is Element of the carrier of b1;

:: LATSUM_1:th 13
theorem
for b1, b2 being non empty RelStr
for b3 being set
      st b3 in (the carrier of b1) /\ the carrier of b2
   holds b3 is Element of the carrier of b2;

:: LATSUM_1:th 14
theorem
for b1, b2 being non empty reflexive transitive antisymmetric with_suprema RelStr
for b3, b4 being Element of the carrier of b1 [*] b2
      st b3 in the carrier of b1 & b4 in the carrier of b2 & b1 tolerates b2
   holds    b3 <= b4
   iff
      ex b5 being Element of the carrier of b1 [*] b2 st
         b5 in (the carrier of b1) /\ the carrier of b2 & b3 <= b5 & b5 <= b4;

:: LATSUM_1:th 15
theorem
for b1, b2 being non empty RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
      st b3 = b5 & b4 = b6 & b1 tolerates b2 & b1 is transitive & b2 is transitive
   holds    b3 <= b4
   iff
      b5 <= b6;

:: LATSUM_1:th 16
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema RelStr
for b2 being directed lower Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3 in b2 & b4 in b2
   holds b3 "\/" b4 in b2;

:: LATSUM_1:th 17
theorem
for b1, b2 being RelStr
for b3, b4 being set
      st (the carrier of b1) /\ the carrier of b2 is upper Element of bool the carrier of b1 &
         [b3,b4] in the InternalRel of b1 [*] b2 &
         b3 in the carrier of b2
   holds b4 in the carrier of b2;

:: LATSUM_1:th 18
theorem
for b1, b2 being RelStr
for b3, b4 being Element of the carrier of b1 [*] b2
      st (the carrier of b1) /\ the carrier of b2 is upper Element of bool the carrier of b1 &
         b3 <= b4 &
         b3 in the carrier of b2
   holds b4 in the carrier of b2;

:: LATSUM_1:th 19
theorem
for b1, b2 being non empty reflexive transitive antisymmetric with_suprema RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
      st (the carrier of b1) /\ the carrier of b2 is directed lower Element of bool the carrier of b2 &
         (the carrier of b1) /\ the carrier of b2 is upper Element of bool the carrier of b1 &
         b1 tolerates b2 &
         b3 = b5 &
         b4 = b6
   holds b3 "\/" b4 = b5 "\/" b6;

:: LATSUM_1:th 20
theorem
for b1, b2 being non empty reflexive transitive antisymmetric with_suprema lower-bounded RelStr
      st (the carrier of b1) /\ the carrier of b2 is non empty directed lower Element of bool the carrier of b2
   holds Bottom b2 in the carrier of b1;

:: LATSUM_1:th 21
theorem
for b1, b2 being RelStr
for b3, b4 being set
      st (the carrier of b1) /\ the carrier of b2 is lower Element of bool the carrier of b2 &
         [b3,b4] in the InternalRel of b1 [*] b2 &
         b4 in the carrier of b1
   holds b3 in the carrier of b1;

:: LATSUM_1:th 22
theorem
for b1, b2 being set
for b3, b4 being RelStr
      st [b1,b2] in the InternalRel of b3 [*] b4 &
         (the carrier of b3) /\ the carrier of b4 is upper Element of bool the carrier of b3 &
         (b1 in the carrier of b3 implies not b2 in the carrier of b3) &
         (b1 in the carrier of b4 implies not b2 in the carrier of b4)
   holds b1 in (the carrier of b3) \ the carrier of b4 & b2 in (the carrier of b4) \ the carrier of b3;

:: LATSUM_1:th 23
theorem
for b1, b2 being RelStr
for b3, b4 being Element of the carrier of b1 [*] b2
      st (the carrier of b1) /\ the carrier of b2 is lower Element of bool the carrier of b2 &
         b3 <= b4 &
         b4 in the carrier of b1
   holds b3 in the carrier of b1;

:: LATSUM_1:th 24
theorem
for b1, b2 being RelStr
      st b1 tolerates b2 &
         (the carrier of b1) /\ the carrier of b2 is upper Element of bool the carrier of b1 &
         (the carrier of b1) /\ the carrier of b2 is lower Element of bool the carrier of b2 &
         b1 is transitive &
         b1 is antisymmetric &
         b2 is transitive &
         b2 is antisymmetric
   holds b1 [*] b2 is antisymmetric;

:: LATSUM_1:th 25
theorem
for b1, b2 being RelStr
      st (the carrier of b1) /\ the carrier of b2 is upper Element of bool the carrier of b1 &
         (the carrier of b1) /\ the carrier of b2 is lower Element of bool the carrier of b2 &
         b1 tolerates b2 &
         b1 is transitive &
         b2 is transitive
   holds b1 [*] b2 is transitive;