
theorem Th27:
  for S being non void non empty ManySortedSign holds
  id the carrier of S, id the carrier' of S form_morphism_between S,S
proof
  let S be non void non empty ManySortedSign;
  set f = id the carrier of S, g = id the carrier' of S;
A1: dom the ResultSort of S = the carrier' of S by FUNCT_2:def 1;
  rng the ResultSort of S c= the carrier of S;
  then f*the ResultSort of S = the ResultSort of S by RELAT_1:53;
  hence dom f = the carrier of S & dom g = the carrier' of S &
  rng f c= the carrier of S & rng g c= the carrier' of S &
  f*the ResultSort of S = (the ResultSort of S)*g by A1,RELAT_1:52;
  let o be set, p be Function;
  assume that
A2: o in the carrier' of S and
A3: p = (the Arity of S).o;
A4: g.o = o by A2,FUNCT_1:17;
  p in (the carrier of S)* by A2,A3,FUNCT_2:5;
  then p is FinSequence of the carrier of S by FINSEQ_1:def 11;
  then rng p c= the carrier of S by FINSEQ_1:def 4;
  hence thesis by A3,A4,RELAT_1:53;
end;
