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 defISOA1:
  for X, Y be RealNormSpace,
      x be Point of X, y be Point of Y holds
    (IsoCPNrSP(X,Y)").<*x,y*> = [x,y]
 proof
    let X, Y be RealNormSpace;
    set I = IsoCPNrSP(X,Y);
    set J = I";
    P0: dom I = the carrier of [:X,Y:] by FUNCT_2:def 1;
    for x be Point of X, y be Point of Y holds J.<*x,y*> = [x,y]
    proof
      let x be Point of X, y be Point of Y;
      Q1: I.(x,y) = <*x,y*> by defISO;
      reconsider z = [x,y] as Point of [:X,Y:];
      J.(I.z) = z by P0,FUNCT_1:34;
      hence thesis by Q1;
    end;
  hence thesis;
end;
