
theorem Th51:
  for R,S being Ring for f being Function of R, S st f is
  RingMonomorphism for x being Element of R holds f.x = 0.S iff x = 0.R
proof
  let R,S be Ring;
  let f be Function of R, S;
  assume
A1: f is RingMonomorphism;
  then
A2: f is RingHomomorphism;
  let x be Element of R;
A3: f is one-to-one by A1;
  f.x = 0.S implies x = 0.R
  proof
    assume
A4: f.x = 0.S;
    f.(0.R) = 0.S by A2,Th50;
    hence thesis by A3,A4,FUNCT_2:19;
  end;
  hence thesis by A2,Th50;
end;
