
theorem Th30:
  for S being non empty non void ManySortedSign, f,g being
Function holds f,g form_a_replacement_in S iff (the carrier of S)-indexing f, (
  the carrier' of S)-indexing g form_a_replacement_in S
proof
  let S be non empty non void ManySortedSign;
  let f,g be Function;
  (id the carrier' of S)+*id the carrier' of S = id the carrier' of S;
  then
A1: (id the carrier' of S)+* ((id the carrier' of S)+*(g|the carrier' of S))
  = ((id the carrier' of S)+*(g|the carrier' of S)) by FUNCT_4:14;
  (id the carrier of S)+*id the carrier of S = id the carrier of S;
  then
A2: (id the carrier of S)+* ((id the carrier of S)+*(f|the carrier of S) ) =
  ((id the carrier of S)+*(f|the carrier of S)) by FUNCT_4:14;
  f,g form_a_replacement_in S iff for o1,o2 being OperSymbol of S st ((the
carrier' of S)-indexing g).o1 = ((the carrier' of S)-indexing g).o2 holds ((the
  carrier of S)-indexing f)*the_arity_of o1 = ((the carrier of S)-indexing f)*
the_arity_of o2 & ((the carrier of S)-indexing f).the_result_sort_of o1 = ((the
  carrier of S)-indexing f).the_result_sort_of o2 by Th29;
  hence thesis by A1,A2;
end;
