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

theorem Th41:
  dom f c= the carrier of S & dom g c= the carrier' of S & f,g
form_a_replacement_in S implies (id the carrier of S) +* f, (id the carrier' of
  S) +* g form_morphism_between S, S with-replacement (f,g)
proof
  assume that
A1: dom f c= the carrier of S and
A2: dom g c= the carrier' of S;
A3: (the carrier' of S)-indexing g = (id the carrier' of S) +* g by A2,
RELAT_1:68;
  (the carrier of S)-indexing f = (id the carrier of S) +* f by A1,RELAT_1:68;
  hence thesis by A3,Th40;
end;
