
theorem
  for V being Abelian add-associative right_zeroed right_complementable
  non empty addLoopStr, M,N being Subset of V, v being Element of V st v in N
  holds M + {v} c= M + N
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr;
  let M,N be Subset of V;
  let v be Element of V;
  assume
A1: v in N;
    let x be object;
    assume
A2: x in M + {v};
    then reconsider x as Element of V;
    x in {u1 + v1 where u1,v1 is Element of V : u1 in M & v1 in {v}} by A2,
RUSUB_4:def 9;
    then consider u2,v2 being Element of V such that
A3: x = u2 + v2 and
A4: u2 in M and
A5: v2 in {v};
    x = u2 + v by A3,A5,TARSKI:def 1;
    then
    x in {u1 + v1 where u1,v1 is Element of V : u1 in M & v1 in N} by A1,A4;
    hence thesis by RUSUB_4:def 9;
end;
