
theorem ll:
for R being Ring,
    S being R-homomorphic Ring, T being S-homomorphic R-homomorphic Ring
for f being additive Function of R,S, g being additive Function of S,T
holds PolyHom(g*f) = (PolyHom g) * (PolyHom f)
proof
let R be Ring, S be R-homomorphic Ring, T be R-homomorphic S-homomorphic Ring;
let f be additive Function of R,S, g be additive Function of S,T;
now let o be object;
  assume o in the carrier of Polynom-Ring R;
  then reconsider p = o as Element of the carrier of Polynom-Ring R;
  now let i be Nat;
    A: dom f = the carrier of R by FUNCT_2:def 1;
    C: dom(PolyHom f) = the carrier of Polynom-Ring R by FUNCT_2:def 1;
    thus ((PolyHom(g*f)).p).i
       = (g*f).(p.i) by FIELD_1:def 2
      .= g.(f.(p.i)) by A,FUNCT_1:13
      .= g.( ((PolyHom f).p).i ) by FIELD_1:def 2
      .= ((PolyHom g).((PolyHom f).p)).i by FIELD_1:def 2
      .= (((PolyHom g) * (PolyHom f)).p).i by C,FUNCT_1:13;
    end;
  then (PolyHom(g*f)).p = ((PolyHom g) * (PolyHom f)).p;
  hence (PolyHom(g*f)).o = ((PolyHom g) * (PolyHom f)).o;
  end;
hence thesis;
end;
