
theorem
  for S being non empty non void ManySortedSign for A1,A2,B1,B2 being
  MSAlgebra over S st the MSAlgebra of A1 = the MSAlgebra of B1 & the MSAlgebra
of A2 = the MSAlgebra of B2 & the Sorts of A1 is_transformable_to the Sorts of
  A2 for h being ManySortedFunction of A1,A2 st h is_homomorphism A1,A2 ex h9
  being ManySortedFunction of B1,B2 st h9 = h & h9 is_homomorphism B1,B2
proof
  let S be non empty non void ManySortedSign;
  let A1,A2,B1,B2 be MSAlgebra over S such that
A1: the MSAlgebra of A1 = the MSAlgebra of B1 and
A2: the MSAlgebra of A2 = the MSAlgebra of B2 and
A3: the Sorts of A1 is_transformable_to the Sorts of A2;
  let h be ManySortedFunction of A1,A2 such that
A4: h is_homomorphism A1,A2;
  reconsider h9 = h as ManySortedFunction of B1,B2 by A1,A2;
  take h9;
  thus h9 = h;
  let o be OperSymbol of S;
  assume
A5: Args(o,B1) <> {};
  then
A6: Args(o,B2) <> {} by A1,A2,A3,Th2;
  let x be Element of Args(o,B1);
  reconsider y = x as Element of Args(o,A1) by A1;
  thus (h9.(the_result_sort_of o)).(Den(o,B1).x) = (h.(the_result_sort_of o)).
  (Den(o,A1).y) by A1
    .= Den(o,A2).(h#y) by A1,A4,A5
    .= Den(o,B2).(h9#x) by A1,A2,A5,A6,Th5;
end;
