
theorem Th2:
  for V be RealNormSpace, V1 be SubRealNormSpace of V
  holds LinearTopSpaceNorm V1 is SubSpace of LinearTopSpaceNorm V
  proof
    let V be RealNormSpace, V1 be SubRealNormSpace of V;
    the carrier of (LinearTopSpaceNorm V1) = the carrier of V1
    & the carrier of (LinearTopSpaceNorm V) = the carrier of V
    by NORMSP_2:def 4; then
    the TopStruct of (LinearTopSpaceNorm V1)
    = the TopStruct of (TopSpaceNorm V1)
    & the TopStruct of (LinearTopSpaceNorm V)
      = the TopStruct of (TopSpaceNorm V) by NORMSP_2:def 4;
   hence thesis by Th1,PRE_TOPC:10;
end;
