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 LMNR0:
  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 ||.z.|| = sqrt (||.x.|| ^2 + ||.y.|| ^2)
  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
    consider w be Element of REAL 2 such that
    A1: w = <* ||.x.||,||.y.|| *> & ||.z.|| = |.w.| by PRVECT_3:18;
    w in 2 -tuples_on REAL; then
    A2: ex s be Element of REAL* st s = w & len s = 2;
    w . 1 = ||.x.|| & w . 2 = ||.y.|| by A1,FINSEQ_1:44;
    hence thesis by A1,A2,EUCLID_3:22;
  end;
