
theorem
for R being domRing
for p being Polynomial of R
for n being Nat holds LC(p`^n) = (LC p)|^n
proof
let R be domRing, p be Polynomial of R; let n be Nat;
per cases;
suppose A: n is zero;
LC(p`^0) = LC(1_.(R)) by POLYNOM5:15
        .= LC((1.R)|R) by RING_4:14
        .= 1_R by RING_5:6
        .= (LC p)|^0 by BINOM:8;
hence thesis by A;
end;
suppose A: n is non zero;
defpred P[Nat] means LC(p`^($1)) = (LC p)|^($1);
LC(p`^1) =  LC p by POLYNOM5:16 .= (LC p)|^1 by BINOM:8; then
IA: P[1];
IS: now let k be non zero Nat;
    assume IV: P[k];
    LC(p`^(k+1)) = LC((p`^k) *' p) by POLYNOM5:19
                .= LC(p`^k) * (LC p) by NIVEN:46
                .= ((LC p)|^k) * ((LC p)|^1) by IV,BINOM:8
                .= (LC p)|^(k+1) by BINOM:10;
    hence P[k+1];
    end;
for k being non zero Nat holds P[k] from NAT_1:sch 10(IA,IS);
hence thesis by A;
end;
end;
