
theorem Th3:
  for F being add-associative right_zeroed right_complementable
  left-distributive non empty doubleLoopStr, x being Element of F holds
  (0.F)*x = 0.F
proof
  let F be add-associative right_zeroed right_complementable left-distributive
  non empty doubleLoopStr;
  let x be Element of F;
  (0.F)*x+(0.F) = ((0.F)+(0.F))*x+(0.F) by RLVECT_1:4
    .= ((0.F)+(0.F))*x by RLVECT_1:4
    .= (0.F)*x+(0.F)*x by Def3;
  hence thesis by RLVECT_1:8;
end;
