
theorem Lm1:
for p being Prime
for R being p-characteristic commutative Ring
for a being Element of R, n being Nat holds (n * p) * a = 0.R
proof
let p be Prime, R be p-characteristic commutative Ring;
let a be Element of R, n be Nat;
defpred P[Nat] means (($1)*p) * a = 0.R;
IA: P[0] by BINOM:12;
IS: now let k be Nat;
    assume IV: P[k];
    ((k+1)*p) * a
         = (k*p+p) * a
        .= (k*p) * a + p * a by BINOM:15
        .= 0.R by IV,Lm0;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
