
theorem Th61:
  for V being RealUnitarySpace, W being Subspace of V, u,v being
VECTOR of V st v + W = u + W holds ex v1 being VECTOR of V st v1 in W & v - v1
  = u
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let u,v be VECTOR of V;
  assume v + W = u + W;
  then u in v + W by Th37;
  then consider u1 being VECTOR of V such that
A1: u = v + u1 and
A2: u1 in W;
  take v1 = v - u;
  0.V = (v + u1) - u by A1,RLVECT_1:15
    .= u1 + (v - u) by RLVECT_1:def 3;
  then v1 = - u1 by RLVECT_1:def 10;
  hence v1 in W by A2,Th16;
  thus v - v1 = (v - v) + u by RLVECT_1:29
    .= 0.V + u by RLVECT_1:15
    .= u by RLVECT_1:4;
end;
