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 NORMSP35:
  for X,Y be RealNormSpace,
        x be Point of X,
        y be Point of Y,
        z be Point of [:X,Y:]
    st z = [x,y]
  holds ||.x.|| <= ||.z.|| & ||.y.|| <= ||.z.||
  proof
    let X,Y be RealNormSpace,
          x be Point of X,
          y be Point of Y,
          z be Point of [:X,Y:];
    assume z = [x,y]; then
    A2: ||.z.|| = sqrt(||.x.|| ^2 + ||.y.|| ^2) by LMNR0;
    0 + ||.x.|| ^2 <= ||.x.|| ^2 + ||.y.|| ^2 by XREAL_1:6; then
    sqrt(||.x.|| ^2) <= sqrt (||.x.|| ^2 + ||.y.|| ^2) by SQUARE_1:26;
    hence ||.x.|| <= ||.z.|| by A2,SQUARE_1:22;
    0 + ||.y.|| ^2 <= ||.x.|| ^2 + ||.y.|| ^2 by XREAL_1:6; then
    sqrt(||.y.|| ^2) <= sqrt(||.x.|| ^2 + ||.y.|| ^2) by SQUARE_1:26;
    hence ||.y.|| <= ||.z.|| by A2,SQUARE_1:22;
  end;
