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

theorem
  for E,F,G be non empty addLoopStr 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]
proof
  let E,F,G be non empty addLoopStr;
  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_3:7;
   consider x1 be Point of E, x2 be Point of F such that
     A2: x1x2=[x1,x2] by PRVECT_3:7;
   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_3:7;
 hence [x1,x2,x3]+[y1,y2,y3] = [x1+y1,x2+y2,x3+y3] by PRVECT_3:7;
end;
thus 0.[:E,F,G:] = [0.E,0.F,0.G];
end;
