reserve AFV for WeakAffVect;
reserve a,b,c,d,e,f,a9,b9,c9,d9,f9,p,q,r,o,x99 for Element of AFV;
reserve a,b,c for Element of GroupVect(AFV,o);
reserve a,b for Element of GroupVect(AFV,o);
reserve AFV for AffVect,
  o for Element of AFV;

theorem Th47:
  for a being Element of GroupVect(AFV,o) st a + a = 0.(GroupVect(
  AFV,o)) holds a = 0.(GroupVect(AFV,o))
proof
  let a be Element of GroupVect(AFV,o) such that
A1: a + a = 0.(GroupVect(AFV,o));
  reconsider a99=a as Element of AFV;
  o = Padd(o,a99,a99) by A1,Def6;
  then
A2: o,a99 // a99,o by Def5;
  o,o // o,o by Th1;
  hence thesis by A2,TDGROUP:16;
end;
