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

theorem Th48:
  for S1, S2 being non empty Signature holds S1+*S2 is Extension of S2
proof
  let S1,S2 be non empty Signature;
  set S = S1+*S2;
  set f1 = id the carrier of S2, g1 = id the carrier' of S2;
  thus dom f1 = the carrier of S2 & dom g1 = the carrier' of S2;
A1: the carrier of S = (the carrier of S1) \/ the carrier of S2 by
CIRCCOMB:def 2;
A2: the carrier' of S = (the carrier' of S1) \/ the carrier' of S2 by
CIRCCOMB:def 2;
  thus rng f1 c= the carrier of S & rng g1 c= the carrier' of S by A1,A2,
XBOOLE_1:7;
A3: the ResultSort of S = (the ResultSort of S1)+*the ResultSort of S2 by
CIRCCOMB:def 2;
  dom the ResultSort of S2 = the carrier' of S2 by FUNCT_2:def 1;
  then the ResultSort of S2 = (the ResultSort of S)|the carrier' of S2 by A3;
  then
A4: the ResultSort of S2 = (the ResultSort of S)*g1 by RELAT_1:65;
  rng the ResultSort of S2 c= the carrier of S2;
  hence f1*the ResultSort of S2 = (the ResultSort of S)*g1 by A4,RELAT_1:53;
  let o be set, p be Function such that
A5: o in the carrier' of S2 and
A6: p = (the Arity of S2).o;
A7: dom the Arity of S2 = the carrier' of S2 by FUNCT_2:def 1;
  then p in rng the Arity of S2 by A5,A6,FUNCT_1:def 3;
  then p is FinSequence of the carrier of S2 by FINSEQ_1:def 11;
  then rng p c= the carrier of S2 by FINSEQ_1:def 4;
  hence f1*p = p by RELAT_1:53
    .= ((the Arity of S1)+*the Arity of S2).o by A5,A6,A7,FUNCT_4:13
    .= (the Arity of S).o by CIRCCOMB:def 2
    .= (the Arity of S).(g1.o) by A5,FUNCT_1:18;
end;
