reserve E, F, G,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 Th17:
  for X, Y be RealNormSpace
  for x be Point of X
  for y be Point of Y
  for z be Point of [:X,Y:] st z = [x,y]
  holds ||.z.|| <= ||.x.|| + ||.y.||
proof
  let X, Y be RealNormSpace;
  let x be Point of X;
  let y be Point of Y;
  let z be Point of [:X,Y:];
  assume z = [x,y];
  then A1: ||.z.|| = sqrt((||.x.|| ^2) + (||.y.|| ^2)) by NDIFF_8:1;

    |. ||.x.|| .| = ||.x.||
  & |. ||.y.|| .| = ||.y.|| by COMPLEX1:43,NORMSP_1:4;
  hence thesis by A1,COMPLEX1:78;
end;
