reserve S for non void non empty ManySortedSign,
  U1,U2 for MSAlgebra over S,
  o for OperSymbol of S,
  n for Nat;

theorem Th2:
  for I be set, A,B,C be ManySortedSet of I, F be
ManySortedFunction of A,B, G be ManySortedFunction of B,C holds dom (G ** F) =
  I & for i be set st i in I holds (G**F).i = (G.i)*(F.i)
proof
  let I be set, A,B,C be ManySortedSet of I, F be ManySortedFunction of A,B, G
  be ManySortedFunction of B,C;
  dom F = I & dom G = I by PARTFUN1:def 2;
  then (dom F) /\ (dom G) = I;
  hence
A1: dom (G ** F) = I by PBOOLE:def 19;
  let i be set;
  thus thesis by A1,PBOOLE:def 19;
end;
