
theorem lemNor3d:
for F1 being Field,
    F2 being F1-homomorphic Field
for h being Homomorphism of F1,F2
for a being Element of F1 holds (PolyHom h).(X-a) = X-(h.a)
proof
let F1 be Field, F2 be F1-homomorphic Field;
let h be Homomorphism of F1,F2; let a be Element of F1;
H: X-a = rpoly(1,a) & X-(h.a) = rpoly(1,h.a)by FIELD_9:def 2;
now let i be Element of NAT;
  per cases;
  suppose A: i = 0;
    set g1 = power(F1), g2 = power(F2);
    thus ((PolyHom h).(X-a)).i
        = h.((X-a).i) by FIELD_1:def 2
       .= h.(-g1.(a,0+1)) by A,H,HURWITZ:25
       .= h.(-(g1.(a,0) * a)) by GROUP_1:def 7
       .= h.(-(1_F1 * a)) by GROUP_1:def 7
       .= -(1_F2 * h.a) by RING_2:7
       .= -(g2.(h.a,0) * h.a) by GROUP_1:def 7
       .= -g2.(h.a,0+1) by GROUP_1:def 7
       .= (X-(h.a)).i by A,H,HURWITZ:25;
     end;
  suppose A: i = 1;
    thus ((PolyHom h).(X-a)).i
        = h.((X-a).i) by FIELD_1:def 2
       .= h.(1_F1) by A,H,HURWITZ:25
       .= 1_F2 by GROUP_1:def 13
       .= (X-(h.a)).i by A,H,HURWITZ:25;
     end;
  suppose A: i <> 0 & i <> 1;
    thus ((PolyHom h).(X-a)).i
        = h.((X-a).i) by FIELD_1:def 2
       .= h.(0.F1) by A,H,HURWITZ:26
       .= 0.F2 by RING_2:6
       .= (X-(h.a)).i by A,H,HURWITZ:26;
     end;
   end;
hence thesis;
end;
