 reserve n for Nat;

theorem
   for R being domRing, S being R-homomorphic domRing
   for h being Homomorphism of R,S
   for p being Element of the carrier of (Polynom-Ring R),
   b,x being Element of R holds
   h.eval(b*p,x) = h.b * eval((PolyHom h).p,h.x)
   proof
     let R be domRing, S be R-homomorphic domRing;
     let h be Homomorphism of R,S;
     let p be Element of the carrier of (Polynom-Ring R), b,x be Element of R;
     reconsider q = b * p as Element of the carrier of Polynom-Ring R
     by POLYNOM3:def 10;
     h.eval(b*p,x) = eval((PolyHom h).q, h.x) by Th28
             .= eval(h.b * (PolyHom h).p,h.x) by Th27
             .= h.b * eval((PolyHom h).p,h.x) by RING_5:7;
     hence thesis;
   end;
