 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 LM0:
  for E,F be RealNormSpace,
      z be Point of [:E,F:],
      x be Point of E,
      y be Point of F
  st z = [x,y]
  holds ||.z.|| <= ||.x.|| + ||.y.||
  proof
    let E,F be RealNormSpace,
        z be Point of [:E,F:],
        x be Point of E,
        y be Point of F;
    assume z = [x,y]; then
    A2: ||.z.|| =sqrt (||.x.|| ^2 + ||.y.|| ^2) by NDIFF_8:1;
    ||.x.|| ^2 + ||.y.|| ^2 + 0
      <= ( ||.x.|| ^2 + ||.y.|| ^2 ) + ( 2 * ||.x.||) * ||.y.||
      by XREAL_1:7; then
    sqrt (||.x.|| ^2 + ||.y.|| ^2 )
     <= sqrt (( ||.x.|| + ||.y.|| ) ^2 ) by SQUARE_1:26;
    hence thesis by A2,SQUARE_1:22;
  end;
