reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
  for X,Y be RealNormSpace,
        y be Point of Y,
        z be Point of [:X,Y:]
  st z = [0.X,y]
  holds ||.z.|| = ||.y.||
  proof
    let X,Y be RealNormSpace,
          y be Point of Y,
          z be Point of [:X,Y:];
    assume z = [0.X,y]; then
    ||.z.|| = sqrt(||.0.X.|| ^2 + ||.y.|| ^2) by LMNR0
           .= sqrt(||.y.|| ^2);
    hence ||.y.|| = ||.z.|| by SQUARE_1:22;
  end;
