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

theorem Th43:
  f,g form_a_replacement_in S implies S with-replacement ((the
  carrier of S)-indexing f, g) = S with-replacement (f,g)
proof
  set X = the carrier of S, Y = the carrier' of S;
  set S2 = S with-replacement (X-indexing f, g);
A1: X-indexing (X-indexing f) = X-indexing f by Th11;
  assume
A2: f,g form_a_replacement_in S;
  then X-indexing f, Y-indexing g form_a_replacement_in S by Th30;
  then
A3: X-indexing f, g form_a_replacement_in S by A1,Th30;
  then
A4: the carrier of S2 = rng (X-indexing f) by A1,Def4;
A5: the carrier' of S2 = rng (Y-indexing g) by A3,Def4;
  X-indexing f, Y-indexing g form_morphism_between S, S2 by A1,A3,Def4;
  hence thesis by A2,A4,A5,Def4;
end;
