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

theorem Th14:
  for E,F,G be RealNormSpace holds
  ( for x be set holds
  ( x is Point of [: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.[: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, 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] = [-x1,-x2,-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;
   thus for x be set holds (x is Point of [: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 x be set;
   hereby assume x is Point of [:E,F,G:]; then
    consider x1x2 be Point of [:E,F:], x3 be Point of G such that
     A1: x=[x1x2,x3] by PRVECT_3:18;
   consider x1 be Point of E, x2 be Point of F such that
     A2: x1x2=[x1,x2] by PRVECT_3:18;
   take x1,x2,x3;
   thus x=[x1,x2,x3] by A1,A2;
  end;
  thus thesis;
end;
hereby
 let x1,y1 be Point of E, x2,y2 be Point of F, x3,y3 be Point of G;
 [x1,x2]+[y1,y2] =[x1+y1,x2+y2] by PRVECT_3:18;
 hence [x1,x2,x3]+[y1,y2,y3] = [x1+y1,x2+y2,x3+y3] by PRVECT_3:18;
end;
thus 0.[:E,F,G:] = [0.E,0.F,0.G];
thus A7: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]
proof
let x1 be Point of E, x2 be Point of F,x3 be Point of G, a be Real;
 thus a*[x1,x2,x3]=[a*[x1,x2],a*x3] by PRVECT_3:18
       .=[a*x1,a*x2,a*x3] by PRVECT_3:18;
end;
hereby
 let x1 be Point of E, x2 be Point of F, x3 be Point of G;
  thus -[x1,x2,x3] = (-1)*[x1,x2,x3] by RLVECT_1:16
          .=[(-1)*x1,(-1)*x2,(-1)*x3 ] by A7
          .=[-x1,(-1)*x2,(-1)*x3 ] by RLVECT_1:16
          .=[-x1,-x2,(-1)*x3 ] by RLVECT_1:16
          .= [-x1,-x2,-x3] by RLVECT_1:16;
end;
  let x1 be Point of E, y1 be Point of F, z1 be Point of G;
  consider v10 be Element of REAL 2 such that
A11: v10=<* ||.[x1,y1].||,||.z1.|| *>
     & prod_NORM([:E,F:],G).([x1,y1],z1) = |.v10.| by PRVECT_3:def 6;
     consider v20 be Element of REAL 2 such that
A12: v20=<* ||.x1.||,||.y1.|| *>
     & prod_NORM(E,F).(x1,y1) = |.v20.| by PRVECT_3:def 6;
    reconsider v1=<* ||.x1.||,||.y1.||,||.z1.|| *>
      as Element of REAL 3 by FINSEQ_2:104;
A14: 0 <= Sum (sqr v20) by RVSUM_1:86;
A15: ||.[x1,y1].||^2 = Sum (sqr v20) by A14,SQUARE_1:def 2,A12
             .= Sum (<*||.x1.||^2,||.y1.||^2 *> ) by A12,TOPREAL6:11
             .= ||.x1.||^2 + ||.y1.||^2 by RVSUM_1:77;
A16: Sum (sqr v10) = Sum ( <*||.[x1,y1].||^2,||.z1.||^2 *> ) by TOPREAL6:11,A11
       .= ||.x1.||^2 + ||.y1.||^2 + ||.z1.||^2 by A15,RVSUM_1:77;
     |.v10.| =|.v1.| by A16,BORSUK_7:17;
     hence thesis by A11,A16;
  end;
