
theorem
  for V be RealNormSpace,
      W be SubRealNormSpace of V,
      V1 be Subset of V
  st the carrier of W = V1
  holds NLin(V1) = the NORMSTR of W
  proof
    let V be RealNormSpace,
        W be SubRealNormSpace of V,
        V1 be Subset of V;
    assume
    A1: the carrier of W = V1;
    set l = NLin(V1);
    A2: the carrier of l c= the carrier of V by RLSUB_1:def 2;
    A3: the carrier of l = the carrier of W by A1,LCL1,RLSUB134;
    A4: 0.l = 0.V by RLSUB_1:def 2;
    A5: the addF of l = (the addF of V)|| the carrier of l by RLSUB_1:def 2;
    A6: the Mult of l = (the Mult of V) | [:REAL, the carrier of l:]
                        by RLSUB_1:def 2;
    A7: 0.W = 0.l by A4,DUALSP01:def 16;
    A8: the addF of W = the addF of l by A3,A5,DUALSP01:def 16;
    the normF of W = (the normF of V) | (the carrier of W) by DUALSP01:def 16
                  .= the normF of l by A2,A3,DefNorm;
    hence thesis by A6,A7,A8,DUALSP01:def 16;
  end;
