
theorem hcon:
for R being Ring,
    S being R-homomorphic Ring
for h being Homomorphism of R,S
for a being Element of R holds (PolyHom h).(a|R) = (h.a)|S
proof
let R be Ring, S be R-homomorphic Ring;
let h be Homomorphism of R,S; let a be Element of R;
reconsider g = a|R as Element of the carrier of Polynom-Ring R
  by POLYNOM3:def 10;
now let i be Element of NAT;
  per cases;
  suppose A: i <> 0;
    thus ((PolyHom h).g).i = h.(g.i) by FIELD_1:def 2
                          .= h.(0.R) by A,poly0
                          .= 0.S by RING_2:6
                          .= ((h.a)|S).i by A,poly0;
    end;
  suppose B: i = 0;
    thus ((PolyHom h).g).i = h.(g.i) by FIELD_1:def 2
                          .= h.a by B,poly0
                          .= ((h.a)|S).i by B,poly0;
    end;
  end;
hence thesis by FUNCT_2:63;
end;
