 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th19:
for X being NormedLinearTopSpace,
    RNS being RealNormSpace
  st RNS = the NORMSTR of X
for x,y be Point of X,
    x1,y1 be Point of RNS,
    a be Real
    st x1=x & y1=y
holds
   x+y = x1+y1
   &
   a*x = a*x1
   &
   x-y = x1-y1
   &
   ||.x.|| = ||.x1.||
proof
let X be NormedLinearTopSpace;
let RNS be RealNormSpace;
 assume A1: RNS = the NORMSTR of X;
let x,y be Point of X,
    x1,y1 be Point of RNS,
    a be Real;
 assume A2:x1=x & y1=y;
  hence x+y =x1+y1 by A1;
  thus a*x =a*x1 by A1,A2;
  thus x-y = x + (-1)*y by RLVECT_1:16
          .= x1 + (-1)*y1 by A1,A2
          .= x1-y1 by RLVECT_1:16;
  thus ||.x.|| = ||.x1.|| by A1,A2;
end;
