
theorem Th1:
  for V be RealNormSpace,
      W be Subspace of V holds
    RSubNormSpace(W) is SubRealNormSpace of V
proof
  let V be RealNormSpace,
      W be Subspace of V;
  set X = RSubNormSpace(W);
  the RLSStruct of X = the RLSStruct of W &
    the normF of X = (the normF of V) | (the carrier of X)
      by Def1; then
  the carrier of X = the carrier of W
   & 0.X = 0. W
   & the addF of X = the addF of W
   & the Mult of X = the Mult of W; then
  the carrier of X c= the carrier of V
   & 0. X = 0. V
   & the addF of X = (the addF of V) || the carrier of X
   & the Mult of X = (the Mult of V) | [:REAL, the carrier of X:]
   & the normF of X = (the normF of V) | the carrier of X
     by RLSUB_1:def 2,Def1;
  hence thesis by DUALSP01:def 16;
end;
