reserve r,s,t,u for Real;

theorem Th5:
  for X being add-associative right_zeroed right_complementable
  non empty addLoopStr, M being Subset of X holds 0.X+M = M
proof
  let X be add-associative right_zeroed right_complementable non empty
  addLoopStr, M be Subset of X;
A1: 0.X+M = {0.X + u where u is Point of X: u in M} by RUSUB_4:def 8;
  thus 0.X+M c= M
  proof
    let x be object;
    assume x in 0.X+M;
    then ex u being Point of X st x = 0.X+u & u in M by A1;
    hence thesis;
  end;
  let x be object;
  assume
A2: x in M;
  then reconsider x as Point of X;
  0.X+x = x;
  hence thesis by A1,A2;
end;
