
theorem Th18:
  for V being RealUnitarySpace, W being Subspace of V holds (0).V
  /\ W = (0).V & W /\ (0).V = (0).V
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  0.V in W by RUSUB_1:11;
  then 0.V in the carrier of W by STRUCT_0:def 5;
  then {0.V} c= the carrier of W by ZFMISC_1:31;
  then
A1: {0.V} /\ (the carrier of W) = {0.V} by XBOOLE_1:28;
  the carrier of (0).V /\ W = (the carrier of (0).V) /\ (the carrier of W)
  by Def2
    .= {0.V} /\ (the carrier of W) by RUSUB_1:def 2;
  hence (0).V /\ W = (0).V by A1,RUSUB_1:def 2;
  hence thesis by Th14;
end;
