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);

theorem Th44:
  for a being Element of GroupVect(AFV,o), a9 being Element of AFV
  st a=a9 holds -a = (Pcom(o)).a9
proof
  let a be Element of GroupVect(AFV,o), a9 be Element of AFV;
  assume
A1: a=a9;
  reconsider aa = (Pcom(o)).a9 as Element of GroupVect(AFV,o);
  Pcom(o,o) = o & o,a9 // Pcom(o,a9),Pcom(o,o) by Th28,Th32;
  then
A2: Padd(o,a9,Pcom(o,a9)) = o by Def5;
  a + aa = (Padd(o)).(a,(Pcom(o,a9))) by Def7
    .= 0.GroupVect(AFV,o) by A1,A2,Def6;
  hence thesis by RLVECT_1:def 10;
end;
