
theorem
  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 consider v1 being VECTOR of V such that
A1: u = v + v1 and
A2: v1 in W;
    take x = - v1;
    thus x in W by A2,Th16;
    thus thesis by A1,RLVECT_1:17;
  end;
  given v1 being VECTOR of V such that
A3: v1 in W and
A4: u = v - v1;
  - v1 in W by A3,Th16;
  hence thesis by A4;
end;
