
theorem t3:
for R being commutative Ring
for p being Polynomial of R
for n being non zero Nat holds (p `^ n).0 = (p.0) |^ n
proof
let R be commutative Ring, p be Polynomial of R; let n be non zero Nat;
defpred P[Nat] means (p `^ ($1)).0 = (p.0) |^ ($1);
    (p`^1).0 = p.0 by POLYNOM5:16 .= (p.0)|^1 by BINOM:8; then
IA: P[1];
IS: now let k be Nat;
    assume k >= 1;
    assume IV: P[k];
    (p`^(k+1)).0
        = ((p`^k) *' p).0 by POLYNOM5:19
       .= (p`^k).0 * (p.0) by t3a
       .= (p.0)|^k * (p.0)|^1 by IV,BINOM:8
       .= (p.0)|^ (k+1) by BINOM:10;
    hence P[k+1];
    end;
I: for k being Nat st k >= 1 holds P[k] from NAT_1:sch 8(IA,IS);
n >= 0 + 1 by INT_1:7;
hence thesis by I;
end;
