reserve a, I for set,
  S for non empty non void ManySortedSign;
reserve A, M for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem Th25:
  for A being non-empty MSAlgebra over S for F being
  ManySortedFunction of A, Trivial_Algebra S holds F is_epimorphism A,
  Trivial_Algebra S
proof
  let A be non-empty MSAlgebra over S, F be ManySortedFunction of A,
  Trivial_Algebra S;
  set I = the carrier of S;
  consider XX being ManySortedSet of I such that
A1: {XX} = I --> {0} by Th5;
  thus F is_homomorphism A, Trivial_Algebra S
  proof
    let o be OperSymbol of S such that
    Args(o,A) <> {};
    let x be Element of Args(o,A);
    thus (F.the_result_sort_of o).(Den(o,A).x) = 0 by Th24
      .= Den(o,Trivial_Algebra S).(F#x) by Th24;
  end;
  the Sorts of Trivial_Algebra S = {XX} by A1,MSAFREE2:def 12;
  hence thesis by Th9;
end;
