reserve G,F for RealLinearSpace;

theorem Th15:
  for X,Y be RealNormSpace
  holds ex I be Function of [:X,Y:],product <*X,Y*>
  st I is one-to-one & I is onto
  & ( for x be Point of X, y be Point of Y holds I.(x,y) = <*x,y*> )
  & ( for v,w be Point of [:X,Y:] holds I.(v+w) = I.v + I.w )
  & ( for v be Point of [:X,Y:], r be Real
  holds I.(r*v)=r*(I.v) )
  & 0. product <*X,Y*> = I.(0.[:X,Y:])
  & ( for v be Point of [:X,Y:] holds ||. I.v .|| = ||.v.|| )
  proof
    let X,Y be RealNormSpace;
    reconsider X0=X, Y0=Y as RealLinearSpace;
    consider I0 be Function of [:X0,Y0:],product <*X0,Y0*> such that
    A1: I0 is one-to-one & I0 is onto
    & ( for x be Point of X, y be Point of Y holds I0.(x,y) = <*x,y*> )
    & ( for v,w be Point of [:X0,Y0:] holds I0.(v+w) = I0.v + I0.w )
    & ( for v be Point of [:X0,Y0:], r be Real
    holds I0.(r*v)=r*(I0.v) )
    & 0. product <*X0,Y0*>  = I0.(0.[:X0,Y0:]) by Th12;
    A2:product <*X,Y*>
    = NORMSTR(# product carr <*X,Y*>, zeros <*X,Y*>, [:addop <*X,Y*>:],
    [:multop <*X,Y*>:], productnorm <*X,Y*> #) by PRVECT_2:6; then
    reconsider I = I0 as Function of [:X,Y:],product <*X,Y*>;
    take I;
    thus I is one-to-one & I is onto
    & ( for x be Point of X,y be Point of Y
    holds I.(x,y) = <*x,y*> ) by A1,A2;
    thus for v,w be Point of [:X,Y:] holds I.(v+w) = I.v + I.w
    proof
      let v,w be Point of [:X,Y:];
      reconsider v0=v, w0=w as Point of [:X0,Y0:];
      thus I.(v+w) = I0.(v0+w0)
      .= I0.v0 + I0.w0 by A1
      .= I.v + I.w by A2;
    end;
    thus for  v be Point of [:X,Y:], r be Real
    holds I.(r*v)=r*(I.v)
    proof
      let v be Point of [:X,Y:], r be Real;
      reconsider v0=v as Point of [:X0,Y0:];
      thus I.(r*v) = I0.(r*v0)
      .= r*(I0.v0) by A1
      .= r*(I.v) by A2;
    end;
    thus 0. product <*X,Y*> = I.(0.[:X,Y:]) by A1,A2;
    for v be Point of [:X,Y:] holds ||. I.v .|| = ||.v.||
    proof
      let v be Point of [:X,Y:];
      consider x1 be Point of X, y1 be Point of Y such that
      A3: v = [x1,y1] by Lm1;
      consider v1 be Element of REAL 2 such that
      A4: v1=<* ||.x1.||,||.y1.|| *> & prod_NORM(X,Y).(x1,y1) = |.v1.|
        by Def6;
      A5:I.v = I.(x1,y1) by A3
      .= <*x1,y1*> by A1;
      reconsider Iv=I.v as Element of product carr <*X,Y*> by A2;
      1 in {1,2} & 2 in {1,2} by TARSKI:def 2; then
      reconsider j1=1, j2=2 as Element of dom <*X,Y*>
        by FINSEQ_1:2,89;
      A7: normsequence(<*X,Y*>,Iv).j1
      = (the normF of <*X,Y*>.j1).(Iv.j1) by PRVECT_2:def 11
      .= ||.x1.|| by A5;
      A8: normsequence(<*X,Y *>,Iv).j2
      = (the normF of <*X,Y*>.j2).(Iv.j2) by PRVECT_2:def 11
      .= ||.y1.|| by A5;
      len normsequence(<*X,Y*>,Iv)
      = len <*X,Y*> by PRVECT_2:def 11
      .= 2 by FINSEQ_1:44; then
      normsequence(<*X,Y*>,Iv) = v1 by A4,A7,A8,FINSEQ_1:44;
      hence thesis by A4,A3,A2,PRVECT_2:def 12;
    end;
    hence thesis;
  end;
