reserve S,S9 for non void Signature,
  f,g for Function;

theorem
  dom f = the carrier of S & dom g = the carrier' of S & f,g
form_a_replacement_in S implies f,g form_morphism_between S, S with-replacement
  (f,g)
proof
  assume that
A1: dom f = the carrier of S and
A2: dom g = the carrier' of S;
  dom g = dom id the carrier' of S by A2;
  then
A3: (id the carrier' of S) +* g = g by FUNCT_4:19;
  dom f = dom id the carrier of S by A1;
  then (id the carrier of S) +* f = f by FUNCT_4:19;
  hence thesis by A1,A2,A3,Th41;
end;
