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

theorem
  for X, Y be RealLinearSpace holds
    (IsoCPRLSP(X,Y)").(0. product <*X,Y*>) = 0.[:X,Y:]
  proof
    let X, Y be RealLinearSpace;
    set I = IsoCPRLSP(X,Y);
    set J = I";
    A1: dom I = the carrier of [:X,Y:] by FUNCT_2:def 1;
    J.(0. product <*X,Y*>) = J.(I.(0.[:X,Y:])) by ZeZe;
    hence thesis by A1,FUNCT_1:34;
  end;
