reserve UA for Universal_Algebra;
reserve S for non void non empty ManySortedSign,
  U1 for non-empty MSAlgebra over S;

theorem Th9:
  for F be ManySortedFunction of MSAlg UA, MSAlg UA for f be
  Element of UAEnd UA st F = 0 .--> f holds F in MSAEnd MSAlg UA
proof
  let F be ManySortedFunction of MSAlg UA, MSAlg UA;
  let f be Element of UAEnd UA;
  assume F = 0 .--> f;
  then
A1: F = MSAlg f by MSUHOM_1:def 3;
  f is_homomorphism by Def1;
  then MSAlg f is_homomorphism MSAlg UA, MSAlg UA Over MSSign UA by MSUHOM_1:16
;
  then F is_homomorphism MSAlg UA, MSAlg UA by A1,MSUHOM_1:9;
  hence thesis by Def4;
end;
