
theorem Lm0:
for p being Prime
for R being p-characteristic commutative Ring
for a being Element of R holds p * a = 0.R
proof
let p be Prime, R be p-characteristic commutative Ring, a be Element of R;
B: Char R = p by RING_3:def 6;
consider n being Nat such that
A: (p = n & p '*' a = n*a) or (p = -n & p '*' a = -(n*a)) by RING_3:def 2;
thus p * a
   = p '*' (a * 1.R) by A
  .= (p '*' 1.R) * a by REALALG2:5
  .= 0.R * a by B,RING_3:def 5
  .= 0.R;
end;
