 reserve X,Y,Z,E,F,G,S,T for RealLinearSpace;

theorem ZeZe:
  for X, Y be RealLinearSpace holds
    0.product <*X,Y*> = IsoCPRLSP(X,Y).(0.[:X,Y:])
  proof
    let X, Y be RealLinearSpace;
    consider I be Function of [:X,Y:], product <*X,Y*> such that
    I is one-to-one onto and
    A1: for x be Point of X, y be Point of Y holds I.(x,y) = <*x,y*> and
    A2: for v,w be Point of [:X,Y:] holds I.(v+w) = I.v + I.w and
    A3: for v be Point of [:X,Y:], r be Real holds I.(r*v) = r*(I.v) and
    A4: 0. product <*X,Y*> = I.(0.[:X,Y:]) by PRVECT_3:12;
    reconsider I as LinearOperator of [:X,Y:], product <*X,Y*>
      by A2,A3,LOPBAN_1:def 5,VECTSP_1:def 20;
    for a being Element of X,
        b being Element of Y holds I.(a,b) = IsoCPRLSP(X,Y).(a,b)
          by A1,defISO;
    hence thesis by A4,BINOP_1:2;
  end;
