reserve ADG for Uniquely_Two_Divisible_Group;
reserve a,b,c,d,a9,b9,c9,p,q for Element of ADG;
reserve x,y for set;
reserve AS for non empty AffinStruct;

theorem
  (ex a,b being Element of ADG st a<>b) implies AV(ADG) is AffVect
proof
A1: ( for a,b,c being Element of AV(ADG) st a,b // c,c holds a=b)& for a,b,c
,b9 being Element of AV(ADG) st a,b // b,c & a,b9 // b9,c holds b = b9 by Th15;
  assume
A2: ex a,b being Element of ADG st a<>b;
  then
A3: ( for a,b,c,a9,b9,c9 being Element of AV(ADG) st a,b // a9,b9 & a,c //
a9,c9 holds b,c // b9,c9)& for a,c being Element of AV(ADG) ex b being Element
  of AV (ADG) st a,b // b,c by Th15;
A4: for a,b,c,d being Element of AV(ADG) st a,b // c,d holds a,c // b,d by A2
,Th15;
  ( for a,b,c,d,p,q being Element of AV(ADG) st a,b // p,q & c,d // p,q
holds a,b // c,d)& for a,b,c being Element of AV(ADG) ex d being Element of AV(
  ADG) st a,b // c,d by A2,Th15;
  hence thesis by A2,A3,A1,A4,Def5,STRUCT_0:def 10;
end;
