
theorem Th29:
  for S1,S2 being non empty ManySortedSign for f being Function of
  the carrier of S1, the carrier of S2 for g being Function st f,g
  form_morphism_between S1,S2 for A1,A2 being MSAlgebra over S2 for h being
  ManySortedFunction of the Sorts of A1, the Sorts of A2 holds h*f is
  ManySortedFunction of the Sorts of A1|(S1,f,g), the Sorts of A2|(S1,f,g)
proof
  let S1,S2 be non empty ManySortedSign;
  let f be Function of the carrier of S1, the carrier of S2;
  let g be Function such that
A1: f,g form_morphism_between S1,S2;
  let A1,A2 be MSAlgebra over S2;
  let h be ManySortedFunction of the Sorts of A1,the Sorts of A2;
  set B1 = A1|(S1,f,g), B2 = A2|(S1,f,g);
  let x be object;
  assume x in the carrier of S1;
  then reconsider s = x as SortSymbol of S1;
  reconsider fs = f.s as SortSymbol of S2;
A2: (h*f).s = h.fs & (the Sorts of A1).fs = ((the Sorts of A1)*f).s by
FUNCT_2:15;
A3: (the Sorts of A2).fs = ((the Sorts of A2)*f).s by FUNCT_2:15;
  (the Sorts of A1)*f = the Sorts of B1 & (the Sorts of A2)*f = the Sorts
  of B2 by A1,Def3;
  hence thesis by A2,A3;
end;
