Article EXTENS_1, MML version 4.99.1005

:: EXTENS_1:th 5
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non-empty ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3
for b5 being ManySortedSubset of b2
      st b2 c= b5
   holds b4 || b5 = b4;

:: EXTENS_1:th 6
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
for b4 being ManySortedSubset of b2
for b5 being ManySortedFunction of b2,b3 holds
   b5 .:.: b4 c= b5 .:.: b2;

:: EXTENS_1:th 7
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non-empty ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3
for b5, b6 being ManySortedSubset of b2
      st b5 c= b6
   holds (b4 || b6) .:.: b5 = b4 .:.: b5;

:: EXTENS_1:th 8
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3, b4 being non-empty ManySortedSet of b1
for b5 being ManySortedFunction of b2,b3
for b6 being ManySortedFunction of b3,b4
for b7 being ManySortedSubset of b2 holds
   (b6 ** b5) || b7 = b6 ** (b5 || b7);

:: EXTENS_1:th 9
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
   st b2 is_transformable_to b3
for b4 being ManySortedFunction of b2,b3
for b5 being ManySortedSet of b1
      st b3 is ManySortedSubset of b5
   holds b4 is ManySortedFunction of b2,b5;

:: EXTENS_1:th 10
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non-empty ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3
for b5 being ManySortedSubset of b2
      st b4 is "1-1"
   holds b4 || b5 is "1-1";

:: EXTENS_1:th 11
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non-empty ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3
for b5 being ManySortedSubset of b2 holds
   doms (b4 || b5) c= doms b4;

:: EXTENS_1:th 12
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non-empty ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3
for b5 being ManySortedSubset of b2 holds
   rngs (b4 || b5) c= rngs b4;

:: EXTENS_1:th 13
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3 holds
      b4 is "onto"(b1, b2, b3)
   iff
      rngs b4 = b3;

:: EXTENS_1:th 14
theorem
for b1 being non empty non void ManySortedSign
for b2 being non-empty ManySortedSet of the carrier of b1 holds
   rngs Reverse b2 = b2;

:: EXTENS_1:th 15
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3, b4 being non-empty ManySortedSet of b1
for b5 being ManySortedFunction of b2,b3
for b6 being ManySortedFunction of b3,b4
for b7 being non-empty ManySortedSubset of b3
      st rngs b5 c= b7
   holds (b6 || b7) ** b5 = b6 ** b5;

:: EXTENS_1:th 16
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non-empty ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3 holds
      b4 is "onto"(b1, b2, b3)
   iff
      for b5 being non-empty ManySortedSet of b1
      for b6, b7 being ManySortedFunction of b3,b5
            st b6 ** b4 = b7 ** b4
         holds b6 = b7;

:: EXTENS_1:th 17
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non-empty ManySortedSet of b1
for b4 being ManySortedFunction of b2,b3
      st b2 is non-empty & b3 is non-empty
   holds    b4 is "1-1"
   iff
      for b5 being ManySortedSet of b1
      for b6, b7 being ManySortedFunction of b5,b2
            st b4 ** b6 = b4 ** b7
         holds b6 = b7;

:: EXTENS_1:th 18
theorem
for b1 being non empty non void ManySortedSign
for b2 being non-empty MSAlgebra over b1
for b3 being non-empty ManySortedSet of the carrier of b1
for b4, b5 being ManySortedFunction of the Sorts of FreeMSA b3,the Sorts of b2
      st b4 is_homomorphism FreeMSA b3,b2 & b5 is_homomorphism FreeMSA b3,b2 & b4 || FreeGen b3 = b5 || FreeGen b3
   holds b4 = b5;

:: EXTENS_1:th 19
theorem
for b1 being non empty non void ManySortedSign
for b2, b3 being non-empty MSAlgebra over b1
for b4 being ManySortedFunction of the Sorts of b2,the Sorts of b3
   st b4 is_homomorphism b2,b3 & b4 is_epimorphism b2,b3
for b5 being non-empty MSAlgebra over b1
for b6, b7 being ManySortedFunction of the Sorts of b3,the Sorts of b5
      st b6 is_homomorphism b3,b5 & b7 is_homomorphism b3,b5 & b6 ** b4 = b7 ** b4
   holds b6 = b7;

:: EXTENS_1:th 20
theorem
for b1 being non empty non void ManySortedSign
for b2, b3 being non-empty MSAlgebra over b1
for b4 being ManySortedFunction of the Sorts of b2,the Sorts of b3
      st b4 is_homomorphism b2,b3
   holds    b4 is_monomorphism b2,b3
   iff
      for b5 being non-empty MSAlgebra over b1
      for b6, b7 being ManySortedFunction of the Sorts of b5,the Sorts of b2
            st b6 is_homomorphism b5,b2 & b7 is_homomorphism b5,b2 & b4 ** b6 = b4 ** b7
         holds b6 = b7;

:: EXTENS_1:exreg 1
registration
  let a1 be non empty non void ManySortedSign;
  let a2 be non-empty MSAlgebra over a1;
  cluster Relation-like non-empty Function-like GeneratorSet of a2;
end;

:: EXTENS_1:th 21
theorem
for b1 being non empty non void ManySortedSign
for b2 being MSAlgebra over b1
for b3, b4 being ManySortedSubset of the Sorts of b2
      st b3 is ManySortedSubset of b4
   holds GenMSAlg b3 is MSSubAlgebra of GenMSAlg b4;

:: EXTENS_1:th 22
theorem
for b1 being non empty non void ManySortedSign
for b2 being MSAlgebra over b1
for b3 being MSSubAlgebra of b2
for b4 being ManySortedSubset of the Sorts of b2
for b5 being ManySortedSubset of the Sorts of b3
      st b4 = b5
   holds GenMSAlg b4 = GenMSAlg b5;

:: EXTENS_1:th 23
theorem
for b1 being non empty non void ManySortedSign
for b2 being strict non-empty MSAlgebra over b1
for b3 being non-empty MSAlgebra over b1
for b4 being GeneratorSet of b2
for b5, b6 being ManySortedFunction of the Sorts of b2,the Sorts of b3
      st b5 is_homomorphism b2,b3 & b6 is_homomorphism b2,b3 & b5 || b4 = b6 || b4
   holds b5 = b6;