
theorem Th52:
  for R,S being non degenerated almost_left_invertible commutative
Ring for f being Function of R, S st f is RingHomomorphism for x being Element
  of R st x <> 0.R holds f.(x") = (f.x)"
proof
  let R,S be non degenerated almost_left_invertible commutative Ring;
  let f be Function of R, S;
  assume
A1: f is RingHomomorphism;
  let x be Element of R;
  assume
A2: x <> 0.R;
A3: f.x * f.(x") = f.(x"*x) by A1,GROUP_6:def 6
    .= f.(1_R) by A2,VECTSP_1:def 10
    .= 1_S by A1,GROUP_1:def 13;
  then f.x <> 0.S;
  hence thesis by A3,VECTSP_1:def 10;
end;
