
theorem Th1:
  for R being right_zeroed left_add-cancelable left-distributive
  non empty doubleLoopStr, a being Element of R holds 0.R * a = 0.R
proof
  let R be right_zeroed left_add-cancelable left-distributive non empty
  doubleLoopStr, a be Element of R;
  0.R * a = (0.R + 0.R) * a by RLVECT_1:def 4
    .= 0.R * a + 0.R * a by VECTSP_1:def 3;
  then 0.R * a + 0.R * a = 0.R * a + 0.R by RLVECT_1:def 4;
  hence thesis by ALGSTR_0:def 3;
end;
