
theorem Th11:
  for V being RealUnitarySpace, W being Subspace of V holds
  (Omega).V + W = the UNITSTR of V & W + (Omega).V = the UNITSTR of V
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  the carrier of V = the carrier of the UNITSTR of V & the carrier of W c=
  the carrier of V by RUSUB_1:def 1;
  then the carrier of W c= the carrier of (Omega).V by RUSUB_1:def 3;
  then W + (Omega).V = (Omega).V by Lm3
    .= the UNITSTR of V by RUSUB_1:def 3;
  hence thesis by Lm1;
end;
