reserve v,x,x1,x2,y,z for object,
  X,X1,X2,X3 for set;

theorem
  for E,F,G be RealNormSpace holds
  ( for x be set holds
  ( x is Point of product <*E,F,G*>
  iff ex x1 be Point of E, x2 be Point of F,
    x3 be Point of G st x=<* x1,x2,x3 *> ) )
  & ( for x1,y1 be Point of E, x2,y2 be Point of F, x3,y3 be Point of G
    holds <*x1,x2,x3*> + <*y1,y2,y3*> = <* x1+y1,x2+y2,x3+y3 *> )
  & 0.(product <*E,F,G*>) = <* 0.E,0.F,0.G *>
  & ( for x1 be Point of E, x2 be Point of F, x3 be Point of G
    holds -<*x1,x2,x3*> = <* -x1,-x2,-x3 *> )
  & ( for x1 be Point of E, x2 be Point of F, x3 be Point of G, a be Real
    holds a*<*x1,x2,x3*> = <* a*x1,a*x2,a*x3 *> )
  & ( for x1 be Point of E, x2 be Point of F, x3 be Point of G
   holds
   ||. <*x1,x2,x3*> .|| = sqrt (||.x1.||^2+||.x2.||^2+||.x3.||^2)
  &
  ex w be Element of REAL 3 st
  w=<* ||.x1.||,||.x2.||,||.x3.|| *> & ||. <*x1,x2,x3*> .|| = |.w.| )
  proof
    let E,F,G be RealNormSpace;
    consider I be Function of [:E,F,G:],product <*E,F,G*> such that
    A1: I is one-to-one & I is onto
    & ( for x be Point of E, y be Point of F,
         z be Point of G holds I.(x,y,z) = <*x,y,z*> )
    & ( for v,w be Point of [:E,F,G:] holds I.(v+w) = I.v + I.w )
    & ( for v be Point of [:E,F,G:], r be Real holds I.(r*v)=r*(I.v) )
    & 0. product <*E,F,G*> = I.(0.[:E,F,G:])
    & ( for v be Point of [:E,F,G:] holds ||. I.v .|| = ||.v.|| ) by Th15;
    thus for x be set holds
    (x is Point of product <*E,F,G*>
    iff ex x1 be Point of E, x2 be Point of F,
       x3 be Point of G st x=<*x1,x2,x3*> )
    proof
      let y be set;
      hereby assume y is Point of product <*E,F,G*>; then
        y in the carrier of product <*E,F,G*>; then
        y in rng I by A1,FUNCT_2:def 3; then
        consider x be Element of the carrier of [:E,F,G:] such that
        A3:  y = I.x by FUNCT_2:113;
        consider x1 be Point of E, x2 be Point of F, x3 be Point of G
        such that
        A4:  x=[x1,x2,x3] by Th14;
        take x1,x2,x3;
        I.(x1,x2,x3) = <*x1,x2,x3*> by A1;
        hence y= <*x1,x2,x3*> by A3,A4;
      end;
      thus thesis;
    end;
    thus
    A7: for x1,y1 be Point of E, x2,y2 be Point of F, x3,y3 be Point of G
    holds <*x1,x2,x3*>+<*y1,y2,y3*> = <* x1+y1,x2+y2,x3+y3 *>
    proof
      let x1,y1 be Point of E, x2,y2 be Point of F, x3,y3 be Point of G;
      A9: [x1,x2,x3]+[y1,y2,y3] = [x1+y1,x2+y2,x3+y3] by Th14;
      A10: I.(x1+y1,x2+y2,x3+y3) = <* x1+y1,x2+y2,x3+y3 *> by A1;
      I.(x1,x2,x3) = <* x1,x2,x3 *> & I.(y1,y2,y3) = <* y1,y2,y3 *> by A1;
      hence thesis by A1,A9,A10;
    end;
    thus
    A11: 0. product <*E,F,G*> = <* 0.E,0.F,0.G *>
    proof
      I.(0.E,0.F,0.G) =<* 0.E,0.F,0.G *> by A1;
      hence thesis by A1;
    end;
    hereby
      let x1 be Point of E, x2 be Point of F, x3 be Point of G;
      <*x1,x2,x3*>+<* -x1,-x2,-x3 *> = <* x1+-x1,x2+-x2,x3+-x3 *> by A7
      .= <* 0.E,x2+-x2,x3+-x3 *> by RLVECT_1:def 10
      .= <* 0.E,0.F,x3+-x3 *> by RLVECT_1:def 10
      .= 0.(product <*E,F,G*>) by A11,RLVECT_1:def 10;
      hence -<*x1,x2,x3*> = <* -x1,-x2,-x3 *> by RLVECT_1:def 10;
    end;
    hereby
      let x1 be Point of E, x2 be Point of F, x3 be Point of G, a be Real;
      A14: <*x1,x2,x3*> = I.(x1,x2,x3) by A1;
      I.(a*[x1,x2,x3]) = I.(a*x1,a*x2,a*x3) by Th14
      .= <* a*x1,a*x2,a*x3 *> by A1;
      hence a*<*x1,x2,x3*> = <* a*x1,a*x2,a*x3 *> by A14,A1;
    end;
    let x1 be Point of E, x2 be Point of F, x3 be Point of G;
A16: I.[x1,x2,x3] = I.(x1,x2,x3) .= <*x1,x2,x3*> by A1;
     ||.[x1,x2,x3].|| = sqrt (||.x1.||^2+||.x2.||^2+||.x3.||^2) by Th14;
     hence ||.<*x1,x2,x3*>.|| = sqrt (||.x1.||^2+||.x2.||^2+||.x3.||^2)
     by A1,A16;
     consider w be Element of REAL 3 such that
A17: w=<* ||.x1.||,||.x2.||,||.x3.|| *> & ||.[x1,x2,x3].|| = |.w.| by Th14;
     take w;
     thus w=<* ||.x1.||,||.x2.||,||.x3.|| *> by A17;
     thus||.<*x1,x2,x3*>.|| = |.w.| by A1,A17,A16;
   end;
