reserve G,F for RealLinearSpace;

theorem Th18:
  for G,F be RealNormSpace holds
  ( for x be set holds
  ( x is Point of [:G,F:]
  iff ex x1 be Point of G ,x2 be Point of F st x=[x1,x2]) )
  & ( for x,y be Point of [:G,F:], x1,y1 be Point of G, x2,y2 be Point of F
  st x=[x1,x2] & y=[y1,y2] holds x+y = [x1+y1,x2+y2] )
  & 0.[:G,F:] = [0.G,0.F]
  & ( for x be Point of [:G,F:], x1 be Point of G, x2 be Point of F
  st x=[x1,x2] holds -x = [-x1,-x2] )
  & ( for x be Point of [:G,F:], x1 be Point of G, x2 be Point of F,
  a be Real
  st x=[x1,x2] holds a*x = [a*x1,a*x2] )
  & ( for x be Point of [:G,F:], x1 be Point of G, x2 be Point of F
  st x=[x1,x2] holds
  ex w be Element of REAL 2 st
  w=<* ||.x1.||,||.x2.|| *> & ||.x.|| = |.w.| )
  proof
    let G,F be RealNormSpace;
    thus for x be set holds
    ( x is Point of [:G,F:]
    iff ex x1 be Point of G, x2 be Point of F st x=[x1,x2] ) by Lm1;
    thus for x,y be Point of [:G,F:], x1,y1 be Point of G,x2,y2 be Point of F
    st x=[x1,x2] & y=[y1,y2] holds x+y = [x1+y1,x2+y2] by Def1;
    thus 0.[:G,F:] = [0.G,0.F];
    thus for x be Point of [:G,F:], x1 be Point of G ,x2 be Point of F
    st x=[x1,x2] holds -x = [-x1,-x2]
    proof
      let x be Point of [:G,F:];
      let x1 be Point of G, x2 be Point of F;
      assume A1: x=[x1,x2];
      reconsider y= [-x1,-x2 ] as Point of [:G,F:];
      x+y = [x1+-x1,x2+-x2] by A1,Def1
      .= [0.G,x2+-x2] by RLVECT_1:def 10
      .= 0.[:G,F:] by RLVECT_1:def 10;
      hence thesis by RLVECT_1:def 10;
    end;
    thus for x be Point of [:G,F:], x1 be Point of G,
    x2 be Point of F, a be Real
    st x=[x1,x2] holds a*x = [a*x1,a*x2]
    by Def2;
    thus for x be Point of [:G,F:], x1 be Point of G ,x2 be Point of F
    st x=[x1,x2] holds
    ex w be Element of REAL 2
    st w=<* ||.x1.||,||.x2.|| *> & ||.x.||  = |.w.|
    proof
      let x be Point of [:G,F:], x1 be Point of G, x2 be Point of F;
      assume A2: x=[x1,x2];
      ex w be Element of REAL 2 st
      w=<* ||.x1.||,||.x2.|| *> & prod_NORM(G,F).(x1,x2) = |.w.| by Def6;
      hence thesis by A2;
    end;
  end;
