
theorem
  for S1 being non empty ManySortedSign for S2 being non empty
  Subsignature of S1 for A1,A2 being MSAlgebra over S1 for h being
  ManySortedFunction of the Sorts of A1, the Sorts of A2 holds h|the carrier of
  S2 is ManySortedFunction of the Sorts of A1|S2, the Sorts of A2|S2
proof
  let S1 be non empty ManySortedSign;
  let S2 be non empty Subsignature of S1;
  set f = id the carrier of S2, g = id the carrier' of S2;
  let A1,A2 be MSAlgebra over S1;
  let h be ManySortedFunction of the Sorts of A1, the Sorts of A2;
A1: f,g form_morphism_between S2,S1 by Def2;
  then reconsider f as Function of the carrier of S2, the carrier of S1 by Th9;
  h*f is ManySortedFunction of the Sorts of A1|(S2,f,g), the Sorts of A2|(
  S2,f,g) by A1,Th29;
  hence thesis by RELAT_1:65;
end;
