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 ZeZe:
  for X, Y be RealNormSpace holds
    0.product <*X,Y*> = IsoCPNrSP(X,Y).(0.[:X,Y:])
  proof
    let X, Y be RealNormSpace;
    consider I be Function of [:X,Y:], product <*X,Y*> such that
    I is one-to-one onto and
    A2: for x be Point of X, y be Point of Y holds I.(x,y) = <*x,y*> and
    A3: for v,w be Point of [:X,Y:] holds I.(v+w) = I.v + I.w and
    A4: for v be Point of [:X,Y:], r be Real holds I.(r*v) = r*(I.v) and
    A5: 0. product <*X,Y*> = I.(0.[:X,Y:]) and
    for v be Point of [:X,Y:] holds ||. I.v .|| = ||.v.|| by PRVECT_3:15;
    reconsider I as LinearOperator of [:X,Y:], product <*X,Y*>
      by A3,A4,VECTSP_1:def 20,LOPBAN_1:def 5;
    for a being Element of X,
        b being Element of Y holds I.(a,b) = IsoCPNrSP(X,Y).(a,b)
          by defISO,A2;
    hence thesis by A5,BINOP_1:2;
  end;
