
theorem char4:
for R being domRing
holds Char R = 0 iff
      for a being non zero Element of R,
          n being non zero Nat holds n '*' a <> 0.R
proof
let R be domRing;
hereby assume AS: Char R = 0;
   now given a being non zero Element of R,
                   n being non zero Nat such that H: n '*' a = 0.R;
     0.R = n '*' (1.R * a) by H
        .= (n '*' 1.R) * a by c1;
     then n '*' 1.R = 0.R by VECTSP_2:def 1;
     hence contradiction by AS,RING_3:def 5;
     end;
   hence for a being non zero Element of R,
             n being non zero Nat holds n '*' a <> 0.R;
   end;
 assume AS: for a being non zero Element of R,
                   n being non zero Nat holds n '*' a <> 0.R;
  now
  assume Char R <> 0;
    then H1: CharSet R <> {} by RING_3:78;
    let x be Element of CharSet R;
    x in CharSet R by H1;
    then ex n being positive Nat st x = n & n '*' 1.R = 0.R;
    hence contradiction by AS;
   end;
  hence Char R = 0;
end;
