
theorem Th39:
  for V being RealUnitarySpace, v being VECTOR of V holds v + (0). V = {v}
proof
  let V be RealUnitarySpace;
  let v be VECTOR of V;
  thus v + (0).V c= {v}
  proof
    let x be object;
    assume x in v + (0).V;
    then consider u being VECTOR of V such that
A1: x = v + u and
A2: u in (0).V;
A3: the carrier of (0).V = {0.V} by Def2;
    u in the carrier of (0).V by A2;
    then u = 0.V by A3,TARSKI:def 1;
    then x = v by A1,RLVECT_1:4;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {v};
  then
A4: x = v by TARSKI:def 1;
  0.V in (0).V & v = v + 0.V by Th11,RLVECT_1:4;
  hence thesis by A4;
end;
