
theorem NSUBA:
  for V be RealNormSpace,
      W be SubRealNormSpace of V
  st the carrier of W = the carrier of V
  holds the NORMSTR of W = the NORMSTR of V
  proof
    let V be RealNormSpace,
        W be SubRealNormSpace of V;
    assume
    A1: the carrier of W = the carrier of V;
    A2: 0.W = 0.V by DUALSP01:def 16;
    A3: the addF of W = (the addF of V) || the carrier of W by DUALSP01:def 16
                     .= the addF of V by A1;
    A4: the Mult of W
          = (the Mult of V) | [:REAL, the carrier of W:] by DUALSP01:def 16
         .= the Mult of V by A1;
    the normF of W
          = (the normF of V) | (the carrier of W) by DUALSP01:def 16
         .= the normF of V by A1;
    hence thesis by A2,A3,A4;
  end;
