
theorem
for R being non degenerated comRing
for a,b being Element of R
for n being Nat holds eval((X+a)`^n,b) = (a + b)|^n
proof
let R be non degenerated comRing, a,b be Element of R, n be Nat;
defpred P[Nat] means eval((X+a)`^($1),b) = (a + b)|^($1);
eval((X+a)`^0,b)
   = eval(1_.(R),b) by POLYNOM5:15
  .= 1_R by POLYNOM4:18
  .= (a + b)|^0 by BINOM:8; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    eval((X+a)`^(k+1),b)
        = eval(((X+a)`^k) *' (X+a),b) by POLYNOM5:19
       .= ((a + b)|^k) * eval((X+a),b) by IV,POLYNOM4:24
       .= ((a + b)|^k) * (b - -a) by HURWITZ:29
       .= ((a + b)|^k) * ((a + b)|^1) by BINOM:8
       .= (a + b)|^(k+1) by BINOM:10;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
