reserve G,F for RealLinearSpace;

theorem Th16:
  for X be RealNormSpace
  holds ex I be Function of X ,product <*X*>
  st I is one-to-one & I is onto
  & ( for x be Point of X holds I.x = <*x*> )
  & ( for v,w be Point of X holds I.(v+w) = I.v + I.w )
  & ( for v be Point of X, r be Element of REAL holds I.(r*v)=r*(I.v) )
  & 0. product <*X*> = I.(0.X)
  & ( for v be Point of X holds ||. I.v .|| = ||.v.|| )
  proof
    let X be RealNormSpace;
    reconsider X0= X as RealLinearSpace;
    consider I0 be Function of X0,product <*X0*> such that
    A1: I0 is one-to-one & I0 is onto
    & ( for x be Point of X holds I0.x = <*x*> )
    & ( for v,w be Point of X0 holds I0.(v+w) = I0.v + I0.w )
    & ( for v be Point of X0, r be Element of REAL
    holds I0.(r*v)=r*(I0.v) )
    & 0. product <*X0*> = I0.(0.X0) by Th11;
    A2:product <*X*>
    = NORMSTR(# product carr <*X*>, zeros <*X*>, [:addop <*X*>:]
    ,[:multop <*X*>:], productnorm <*X*> #) by PRVECT_2:6; then
    reconsider I=I0 as Function of X,product <*X*>;
    take I;
    thus I is one-to-one & I is onto
    & ( for x be Point of X holds I.x = <*x*> ) by A1,A2;
    thus for v,w be Point of X holds I.(v+w) = I.v + I.w
    proof
      let v,w be Point of X;
      reconsider v0=v, w0=w as Point of X0;
      thus I.(v+w) = I0.v0 + I0.w0 by A1
      .= I.v + I.w by A2;
    end;
    thus for v be Point of X, r be Element of REAL holds I.(r*v)=r*(I.v)
    proof
      let v be Point of X, r be Element of REAL;
      reconsider v0=v as Point of X0;
      thus I.(r*v) = r*(I0.v0) by A1 .= r*(I.v) by A2;
    end;
    thus 0. product <*X*> = I.(0.X) by A1,A2;
    thus for v be Point of X holds ||. I.v .|| = ||.v.||
    proof
      let v be Point of X;
A3:   len <* ||.v.|| *> = 1 by FINSEQ_1:40;
      reconsider vv = ||.v.|| as Element of REAL;
      reconsider v1=<* vv *> as Element of REAL 1 by FINSEQ_2:92,A3;
      reconsider v2 = ||.v.||^2 as Real;
      A4:|.v1.| = sqrt Sum <*v2*> by RVSUM_1:55
      .= sqrt (||.v.||^2) by RVSUM_1:73
      .= ||.v.|| by NORMSP_1:4,SQUARE_1:22;
      A5:I.v = <* v *> by A1;
      reconsider Iv=I.v as Element of product carr <*X*> by A2;
      1 in {1} by TARSKI:def 1; then
      reconsider j1=1 as Element of dom <*X*> by FINSEQ_1:2,def 8;
      A7: normsequence(<*X*>,Iv).j1
      = (the normF of (<*X*>.j1)).(Iv.j1) by PRVECT_2:def 11
      .= ||.v.|| by A5;
      len normsequence(<*X*>,Iv) = len <*X*> by PRVECT_2:def 11
      .= 1 by FINSEQ_1:40; then
      normsequence(<*X*>,Iv) = v1 by A7,FINSEQ_1:40;
      hence thesis by A4,A2,PRVECT_2:def 12;
    end;
  end;
