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

theorem Th52:
  for S being non void Signature, E being Extension of S st
  f,g form_a_replacement_in E holds f,g form_a_replacement_in S
proof
  let S be non void Signature, E be Extension of S;
  set f9 = (the carrier of E)-indexing f;
  set g9 = (the carrier' of E)-indexing g;
  set T = E with-replacement (f,g);
A1: S is Subsignature of E by Def5;
  then
A2: f9|the carrier of S = (the carrier of S)-indexing f by Th17,INSTALG1:10;
A3: g9|the carrier' of S = (the carrier' of S)-indexing g by A1,Th17,
INSTALG1:10;
  assume f,g form_a_replacement_in E;
  then f9, g9 form_morphism_between E, T by Th40;
  then f9|the carrier of S, g9|the carrier' of S form_a_replacement_in S by A1
,Th31,INSTALG1:18;
  hence thesis by A2,A3,Th30;
end;
