
theorem Th60:
  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 v in u + W by Th37;
  then consider u1 being VECTOR of V such that
A1: v = u + u1 and
A2: u1 in W;
  take v1 = u - v;
  0.V = (u + u1) - v by A1,RLVECT_1:15
    .= u1 + (u - v) by RLVECT_1:def 3;
  then v1 = - u1 by RLVECT_1:def 10;
  hence v1 in W by A2,Th16;
  thus v + v1 = (u + v) - v by RLVECT_1:def 3
    .= u + (v - v) by RLVECT_1:def 3
    .= u + 0.V by RLVECT_1:15
    .= u by RLVECT_1:4;
end;
