
theorem lemrpoly:
for R being domRing
for a being Element of R
for m being non zero Nat holds rpoly(1,a)`^m is Ppoly of R
proof
let R be domRing, a be Element of R, m be non zero Nat;
defpred P[Nat] means rpoly(1,a)`^($1) is Ppoly of R;
IA: P[1]
    proof
    rpoly(1,a)`^1 = rpoly(1,a) by POLYNOM5:16;
    hence thesis by RING_5:51;
    end;
IS: now let j be Nat;
    assume 1 <= j;
    assume IV: P[j];
    rpoly(1,a)`^(j+1) = (rpoly(1,a)`^j) *' rpoly(1,a) &
    rpoly(1,a) is Ppoly of R by POLYNOM5:19,RING_5:51;
    hence P[j+1] by IV,RING_5:52;
    end;
for m being Nat st 1 <= m holds P[m] from NAT_1:sch 8(IA,IS);
hence thesis by NAT_1:14;
end;
