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 LMNR1:
  for X,Y be RealNormSpace,
        x be Point of X,
        z be Point of [:X,Y:]
  st z = [x,0.Y]
  holds ||.z.|| = ||.x.||
  proof
    let X,Y be RealNormSpace,
        x be Point of X,
        z be Point of [:X,Y:];
    assume z = [x,0.Y]; then
    ||.z.|| = sqrt (||.x.|| ^2 + ||.0.Y.|| ^2) by LMNR0
           .= sqrt (||.x.|| ^2);
    hence ||.x.|| = ||.z.|| by SQUARE_1:22;
  end;
