
theorem Th50:
  for R,S being Ring for f being Function of R, S st
  f is RingHomomorphism holds f.(0.R) = 0.S
proof
  let R,S be Ring;
  let f be Function of R, S;
  assume
A1: f is RingHomomorphism;
  f.(0.R) = f.(0.R+0.R) by RLVECT_1:4
    .= f.(0.R) + f.(0.R) by A1,VECTSP_1:def 20;
  then 0.S = (f.(0.R) + f.(0.R)) + (-f.(0.R)) by RLVECT_1:def 10
    .= f.(0.R) + (f.(0.R) + (-f.(0.R))) by RLVECT_1:def 3
    .= f.(0.R) + 0.S by RLVECT_1:def 10
    .= f.(0.R) by RLVECT_1:4;
  hence thesis;
end;
