
theorem div2:
for R being comRing,
    a being Element of R
holds a is Unit of R iff {a}-Ideal = [#] R
proof
let R be comRing, a be Element of R;
set A = {a}-Ideal;
B:now assume a is Unit of R;
  then a divides 1.R by GCD_1:def 20;
  then consider c being Element of R such that
  A1: a * c = 1.R;
  now let x be object;
    now assume x in the carrier of R;
      then reconsider x1 = x as Element of R;
      x = x1 * (c * a) by A1
       .= a * (x1 * c) by GROUP_1:def 3;
      then x in the set of all a*r where r is Element of R;
      hence x in A by IDEAL_1:64;
      end;
    hence x in A iff x in the carrier of R;
    end;
  hence A = [#]R by TARSKI:2;
  end;
now assume A = [#]R;
  then a is unital by div0;
  hence a is Unit of R;
  end;
hence thesis by B;
end;
