
theorem
for R being Ring
for q being Element of the carrier of Polynom-Ring R
for p being Polynomial of R
for n being Nat st p = q holds (n * 1.R) * p = n * q
proof
let R be Ring, q be Element of the carrier of Polynom-Ring R;
let p be Polynomial of R; let n be Nat;
assume AS: p = q;
defpred P[Nat] means
   for q being Element of the carrier of Polynom-Ring R
   for p being Polynomial of R
   st p = q holds (($1) * 1.R) * p = ($1) * q;
IA: P[0]
    proof
    now let q be Element of the carrier of Polynom-Ring R;
        let p be Polynomial of R;
      thus (0 * 1.R) * p
         = 0.R * p by BINOM:12
        .= 0_.(R) by POLYNOM5:26
        .= 0.(Polynom-Ring R) by POLYNOM3:def 10
        .= 0 * q by BINOM:12;
      end;
    hence thesis;
    end;
IS: now let k be Nat;
    assume IV: P[k];
    now let q be Element of the carrier of Polynom-Ring R;
        let p be Polynomial of R;
      assume AS: p = q; then
      A: (k * 1.R) * p = k * q by IV;
      thus ((k+1) * 1.R) * p
         = (k * 1.R + 1 * 1.R) * p by BINOM:15
        .= (k * 1.R + 1.R) * p by BINOM:13
        .= (k * 1.R) * p + 1.R * p by BBD
        .= k * q + q by AS,A,POLYNOM3:def 10
        .= k * q + 1 * q by BINOM:13
        .= (k+1) * q by BINOM:15;
      end;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis by AS;
end;
