
theorem Th57:
  for V being RealUnitarySpace, W being Subspace of V, u,v being
  VECTOR of V holds u in v + W iff ex v1 being VECTOR of V st v1 in W & u = v +
  v1
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let u,v be VECTOR of V;
  thus u in v + W implies ex v1 being VECTOR of V st v1 in W & u = v + v1
  proof
    assume u in v + W;
    then ex v1 being VECTOR of V st u = v + v1 & v1 in W;
    hence thesis;
  end;
  given v1 being VECTOR of V such that
A1: v1 in W & u = v + v1;
  thus thesis by A1;
end;
