
theorem Th1:
  for V be RealNormSpace,V1 be SubRealNormSpace of V
  holds TopSpaceNorm V1 is SubSpace of TopSpaceNorm V
  proof
    let V be RealNormSpace,V1 be SubRealNormSpace of V;
    A1: the carrier of (MetricSpaceNorm V1)
        c= the carrier of (MetricSpaceNorm V) by DUALSP01:def 16;
    for x, y being Point of (MetricSpaceNorm V1) holds
    (the distance of (MetricSpaceNorm V1)) . (x,y)
      = (the distance of (MetricSpaceNorm V)) . (x,y)
    proof
      let x,y being Point of (MetricSpaceNorm V1);
      reconsider x1=x,y1=y as Point of V1;
      reconsider x2=x,y2=y as Point of V by A1;
      -y1 = (-1) * y1 by RLVECT_1:16
         .= (-1) * y2 by NORMSP_3:28
         .= -y2 by RLVECT_1:16; then
      A2: x1-y1 = x2-y2 by NORMSP_3:28;
      thus (the distance of (MetricSpaceNorm V1)) . (x,y)
        = ||.x1-y1.|| by NORMSP_2:def 1
       .= ||.x2-y2.|| by A2,NORMSP_3:28
       .= (the distance of (MetricSpaceNorm V)) . (x,y) by NORMSP_2:def 1;
    end; then
    MetricSpaceNorm V1 is SubSpace of (MetricSpaceNorm V) by A1,TOPMETR:def 1;
    hence TopSpaceNorm V1 is SubSpace of TopSpaceNorm V by TOPMETR:13;
  end;
