 reserve n for Nat;

theorem Th27:
   for R being Ring, S being R-homomorphic Ring
   for h being Homomorphism of R,S
   for p being Element of the carrier of (Polynom-Ring R), b being Element of R
   holds (PolyHom h).(b*p) = h.b * (PolyHom h).p
   proof
     let R be Ring, S be R-homomorphic Ring; let h be Homomorphism of R,S;
     let p be Element of the carrier of (Polynom-Ring R), b be Element of R;
     reconsider q = b * p as Element of the carrier of Polynom-Ring R
       by POLYNOM3:def 10;
     reconsider f = (PolyHom h).q as Element of the carrier of Polynom-Ring S;
     now let i be Element of NAT;
       f.i = h.(q.i) by Def2
       .= h.(b * (p.i)) by POLYNOM5:def 4
       .= h.b * h.(p.i) by GROUP_6:def 6
       .= h.b * ((PolyHom h).p).i by Def2
       .= (h.b * (PolyHom h).p).i by POLYNOM5:def 4;
       hence f.i = (h.b * (PolyHom h).p).i;
     end;
     hence thesis by FUNCT_2:63;
   end;
