reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th33:
  for A being non-empty MSAlgebra over S
  for T being non-empty trivial MSAlgebra over S
  for f being ManySortedFunction of A,T holds
  f is_homomorphism A,T
  proof
    let A be non-empty MSAlgebra over S;
    let T be non-empty trivial MSAlgebra over S;
    let f be ManySortedFunction of A,T;
    let o be OperSymbol of S; assume
    Args(o,A) <> {};
    let x be Element of Args(o,A);
A1: dom the ResultSort of S = the carrier' of S by FUNCT_2:def 1; then
    reconsider a = Den(o,A).x as Element of
    (the Sorts of A).the_result_sort_of o by FUNCT_1:13;
    Den(o,T).(f#x) in Result(o,T); then
    (the Sorts of T).the_result_sort_of o is trivial &
    Den(o,T).(f#x) in (the Sorts of T).the_result_sort_of o &
    (f.(the_result_sort_of o)).a in (the Sorts of T).the_result_sort_of o
    by A1,FUNCT_1:13;
    hence (f.(the_result_sort_of o)).(Den(o,A).x) = Den(o,T).(f#x)
    by ZFMISC_1:def 10;
  end;
