
theorem Th13:
  for S,T being feasible ManySortedSign st the carrier of T c= the
  carrier of S & the Arity of T c= the Arity of S & the ResultSort of T c= the
  ResultSort of S holds T is Subsignature of S
proof
  let S,T be feasible ManySortedSign;
  assume that
A1: the carrier of T c= the carrier of S and
A2: the Arity of T c= the Arity of S and
A3: the ResultSort of T c= the ResultSort of S;
  set f = id the carrier of T, g = id the carrier' of T;
  thus dom f = the carrier of T & dom g = the carrier' of T;
  thus rng f c= the carrier of S by A1;
A4: dom the Arity of T = the carrier' of T by FUNCT_2:def 1;
  dom the Arity of S = the carrier' of S & rng g = the carrier' of T by
FUNCT_2:def 1;
  hence rng g c= the carrier' of S by A2,A4,GRFUNC_1:2;
A5: dom the ResultSort of T = the carrier' of T by Th7;
  rng the ResultSort of T c= the carrier of T by RELAT_1:def 19;
  hence f*the ResultSort of T = the ResultSort of T by RELAT_1:53
    .= (the ResultSort of S)|the carrier' of T by A3,A5,GRFUNC_1:23
    .= (the ResultSort of S)*g by RELAT_1:65;
  let o be set, p be Function;
  assume that
A6: o in the carrier' of T and
A7: p = (the Arity of T).o;
  reconsider q = p as Element of (the carrier of T)* by A6,A7,FUNCT_2:5;
  rng q c= the carrier of T by FINSEQ_1:def 4;
  hence f*p = p by RELAT_1:53
    .= (the Arity of S).o by A2,A4,A6,A7,GRFUNC_1:2
    .= (the Arity of S).(g.o) by A6,FUNCT_1:17;
end;
