
theorem Th33:
  for V being non empty addLoopStr, M being Subset of V, v being
  Element of V holds {v} + M = v + M
proof
  let V be non empty addLoopStr;
  let M be Subset of V;
  let v be Element of V;
  for x being object st x in v + M holds x in {v} + M
  proof
    let x be object;
    assume x in v + M;
    then
A1: ex u being Element of V st x = v + u & u in M;
    v in {v} by TARSKI:def 1;
    hence thesis by A1;
  end;
  then
A2: v + M c= {v} + M;
  for x being object st x in {v} + M holds x in v + M
  proof
    let x be object;
    assume x in {v} + M;
    then consider v1,u1 being Element of V such that
A3: x = v1 + u1 and
A4: v1 in {v} and
A5: u1 in M;
    v1 = v by A4,TARSKI:def 1;
    hence thesis by A3,A5;
  end;
  then {v} + M c= v + M;
  hence thesis by A2;
end;
