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

theorem Th35:
  for S1,S2 being non void Signature for f being Function of the
  carrier of S1, the carrier of S2 for g being Function st f,g
  form_morphism_between S1,S2 holds f**the Arity of S1 = (the Arity of S2)*g
proof
  let S1,S2 be non void Signature;
  let f be Function of the carrier of S1, the carrier of S2;
  let g be Function;
A1: dom the Arity of S2 = the carrier' of S2 by FUNCT_2:def 1;
A2: dom ((f*)*the Arity of S1) = the carrier' of S1 by FUNCT_2:def 1;
  assume
A3: f,g form_morphism_between S1,S2;
  then
A4: dom g = the carrier' of S1;
A5: dom the Arity of S1 = the carrier' of S1 by FUNCT_2:def 1;
A6: now
    let c be object;
    assume
A7: c in the carrier' of S1;
    then
A8: (the Arity of S1).c in rng the Arity of S1 by A5,FUNCT_1:def 3;
    then reconsider
    p = (the Arity of S1).c as FinSequence of the carrier of S1 by
FINSEQ_1:def 11;
    thus (f**the Arity of S1).c = f* .p by A5,A7,FUNCT_1:13
      .= f*p by A8,LANG1:def 13
      .= (the Arity of S2).(g.c) by A3,A7
      .= ((the Arity of S2)*g).c by A4,A7,FUNCT_1:13;
  end;
  rng g c= the carrier' of S2 by A3;
  then dom ((the Arity of S2)*g) = the carrier' of S1 by A4,A1,RELAT_1:27;
  hence thesis by A2,A6;
end;
