
theorem
  for G,F be AbGroup holds
  (for x be set holds x is Element of [:G,F:] iff ex x1 be Element of G,
  x2 be Element of F st x = [x1,x2]) &
  (for x,y be Element of [:G,F:], x1,y1 be Element of G, x2,y2 be Element 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 Element of [:G,F:], x1 be Element of G, x2 be Element of F
  st x = [x1,x2] holds -x = [-x1,-x2])
proof
  let G,F be AbGroup;
  thus for x be set holds (x is Element of [:G,F:] iff ex x1 be Element of G,
  x2 be Element of F st x=[x1,x2]) by SUBSET_1:43;
  thus for x,y be Element of [:G,F:], x1,y1 be Element of G,x2,y2 be Element
  of F st x = [x1,x2] & y=[y1,y2] holds x+y = [x1+y1,x2+y2] by PRVECT_3:def 1;
  thus 0.[:G,F:] = [0.G,0.F];
  thus for x be Element of [:G,F:], x1 be Element of G, x2 be Element of F st
  x = [x1,x2] holds -x = [-x1,-x2]
  proof
    let x be Element of [:G,F:];
    let x1 be Element of G, x2 be Element of F;
    assume
A1: x=[x1,x2];
    reconsider y= [-x1,-x2 ] as Element of [:G,F:];
    x+y = [x1+-x1,x2+-x2] by A1,PRVECT_3:def 1
      .= [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;
end;
