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

theorem defISOA1:
  for X, Y be RealLinearSpace,
      x be Point of X, y be Point of Y holds
    (IsoCPRLSP(X,Y)").<*x,y*> = [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;
     let x be Point of X, y be Point of Y;
 A2: I.(x,y) = <*x,y*> by defISO;
     reconsider z = [x,y] as Point of [:X,Y:];
     J.(I.z) = z by A1,FUNCT_1:34;
     hence thesis by A2;
  end;
