
theorem ch2:
for R being non 2-characteristic domRing
for a being Element of R holds 2 '*' a = 0.R iff a = 0.R
proof
let R be non 2-characteristic domRing;
let a be Element of R;
Char R <> 2 by RING_3:def 6; then
H: 2 '*' 1.R <> 0.R by REALALG2:24;
A: now assume 2 '*' a = 0.R;
   then 0.R = 2 '*' (1.R * a) .= (2 '*' 1.R) * a by REALALG2:5;
   hence a = 0.R by H,VECTSP_2:def 1;
   end;
now assume B: a = 0.R;
  (1 + 1) '*' a = 1 '*' a + 1 '*' a by RING_3:62
               .= 1 '*' a + a by RING_3:60 .= a + a by RING_3:60;
  hence 2 '*' a = 0.R by B;
  end;
hence thesis by A;
end;
