
theorem Lm2a:
for p being Prime
for R being p-characteristic commutative Ring
for a being non zero Element of R
for n being Nat holds p divides n iff n * a = 0.R
proof
let p be Prime, R be p-characteristic commutative Ring;
let a be non zero Element of R, n be Nat;
A: now assume p divides n; then
   ex x being Nat st p * x = n by NAT_D:def 3;
   hence n * a = 0.R by Lm1;
   end;
now assume AS: n * a = 0.R;
   now assume A: not p divides n;
     set i = n div p, k = n mod p;
     B: i * p + k = n & k < p by NAT_D:1,2;
     C: (i * p + k) * a = (i * p) * a + k * a by BINOM:15
                       .= 0.R + k * a by Lm1;
     k <> 0 by B,A,NAT_D:3;
     hence contradiction by AS,B,C,Lm0a;
     end;
   hence p divides n;
   end;
hence thesis by A;
end;
