
theorem XYZ:
for R being non degenerated comRing, S being comRingExtension of R
for a being Element of R, b being Element of S
for n being Element of NAT st a = b holds (X-b)`^n = (X-a)`^n
proof
let R be non degenerated comRing, S be comRingExtension of R,
    a be Element of R, b being Element of S, n be Element of NAT;
assume AS: a = b;
defpred P[Nat] means (X-b)`^($1) = (X-a)`^($1);
(X-b)`^0 = 1_.(S) by POLYNOM5:15
        .= 1_.(R) by FIELD_4:14 .= (X-a)`^0 by POLYNOM5:15; then
IA: P[0];
IS: now let k be Nat;
    assume P[k]; then
    H: (X-a)`^k = (X-b)`^k & (X-b) = (X-a) by AS,FIELD_4:21;
    (X-b)`^(k+1)
        = ((X-b)`^k) *' (X-b) by POLYNOM5:19
       .= ((X-a)`^k) *' (X-a) by H,FIELD_4:17
       .= (X-a)`^(k+1) by POLYNOM5:19;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
