 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);
 reserve S,Z for RealNormSpace;
 reserve T for RealBanachSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th15:
  for T be NormedLinearTopSpace,
      RNS be RealNormSpace st
    RNS = the NORMSTR of T &
    the topology of T = the topology of TopSpaceNorm RNS holds
  distance_by_norm_of RNS = distance_by_norm_of T &
  MetricSpaceNorm RNS = MetricSpaceNorm T &
  TopSpaceNorm T = TopSpaceNorm RNS
proof
  let T be NormedLinearTopSpace, RNS be RealNormSpace;
  assume
A1:  RNS = the NORMSTR of T
    & the topology of T = the topology of TopSpaceNorm RNS;
A2: for x, y being Point of RNS
holds (distance_by_norm_of T) . (x,y) = ||. x - y .||
proof
let x, y being Point of RNS;
reconsider x1=x,y1=y as Point of T by A1;
thus (distance_by_norm_of T) . (x,y) = ||.x1 - y1.|| by NORMSP_2:def 1
 .= ||.x - y.|| by C0SP3:19,A1;
end;
hence distance_by_norm_of RNS
     = distance_by_norm_of T by A1,NORMSP_2:def 1;
thus MetricSpaceNorm RNS = MetricSpaceNorm T by A2,NORMSP_2:def 1,A1;
thus TopSpaceNorm T = TopSpaceNorm RNS by A2,NORMSP_2:def 1,A1;
end;
