
theorem Th23:
  for R,S being add-associative right_zeroed right_complementable
  non empty doubleLoopStr, f being Function of R, S st f is RingHomomorphism
  holds f.(0.R) = 0.S
proof
  let R,S be add-associative right_zeroed right_complementable non empty
  doubleLoopStr;
  let f be Function of R, S;
  assume f is RingHomomorphism;
  then
A1: f is additive;
  f.(0.R)=f.(0.R+0.R) by RLVECT_1:4
    .= f.(0.R)+f.(0.R) by A1;
  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;
