
theorem ch4:
for R being non 2-characteristic domRing
for a being Element of R holds 4 '*' a = 0.R iff a = 0.R
proof
let F be non 2-characteristic domRing; let a be Element of F;
Char F <> 2 by RING_3:def 6; then
I: 2 '*' 1.F <> 0.F by REALALG2:24;
H: now assume 4 '*' a = 0.F;
   then 0.F = (2 * 2) '*' a
           .= 2 '*' (2 '*' (1.F * a)) by RING_3:65
           .= 2 '*' ((2 '*' 1.F) * a) by REALALG2:5
           .= 2 '*' (a * (2 '*' 1.F)) by GROUP_1:def 12
           .= (2 '*' a) * (2 '*' 1.F) by REALALG2:5;
   then 2 '*' a = 0.F by I,VECTSP_2:def 1;
   hence a = 0.F by ch2;
   end;
now assume a = 0.F;
  then B: 2 '*' a = 0.F by ch2;
  (2 + 2) '*' a = 2 '*' a + 2 '*' a by RING_3:62;
  hence 4 '*' a = 0.F by B;
  end;
hence thesis by H;
end;
