
theorem Th11:
  for S being feasible ManySortedSign, T being Subsignature of S
holds the ResultSort of T c= the ResultSort of S & the Arity of T c= the Arity
  of S
proof
  let S be feasible ManySortedSign, T be Subsignature of S;
  set f = id the carrier of T, g = id the carrier' of T;
A1: dom the Arity of T = the carrier' of T by FUNCT_2:def 1;
A2: f, g form_morphism_between T,S by Def2;
A3: now
    let x be object;
A4: rng the Arity of T c= (the carrier of T)* by RELAT_1:def 19;
    assume
A5: x in dom the Arity of T;
    then (the Arity of T).x in rng the Arity of T by FUNCT_1:def 3;
    then reconsider
    p = (the Arity of T).x as Element of (the carrier of T)* by A4;
    g.x = x by A1,A5,FUNCT_1:17;
    then rng p c= the carrier of T & f*p = (the Arity of S).x by A2,A1,A5,
FINSEQ_1:def 4;
    hence (the Arity of T).x = (the Arity of S).x by RELAT_1:53;
  end;
  the ResultSort of T = f*the ResultSort of T by FUNCT_2:17
    .= (the ResultSort of S)*g by A2;
  hence the ResultSort of T c= the ResultSort of S by RELAT_1:50;
  dom the Arity of S = the carrier' of S by FUNCT_2:def 1;
  then dom the Arity of T c= dom the Arity of S by A1,Th10;
  hence thesis by A3,GRFUNC_1:2;
end;
