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;
reserve ADG for Proper_Uniquely_Two_Divisible_Group;
reserve f for Function of the carrier of ADG,the carrier of ADG;

theorem Th52:
  for o9 being Element of ADG, o being Element of AV(ADG) st (for
x being Element of ADG holds f.x = o9+x) & o=o9 holds for a,b being Element of
ADG holds f.(a+b) =(Padd(o)).(f.a,f.b) & f.(0.ADG) = 0.(GroupVect(AV(ADG),o)) &
  f.(-a) = (Pcom(o)).(f.a)
proof
  let o9 be Element of ADG, o be Element of AV(ADG);
  assume that
A1: for x being Element of ADG holds f.x = o9+x and
A2: o=o9;
  let a,b be Element of ADG;
  set a9=f.a,b9=f.b;
A3: AV(ADG) = AffinStruct(#the carrier of ADG,CONGRD(ADG)#) by TDGROUP:def 3;
  then reconsider a99=a9,b99=b9 as Element of AV(ADG);
  thus f.(a+b) =(Padd(o)).((f.a),(f.b))
  proof
A4: ((Padd(o)).((f.a),(f.b))) = Padd(o,a99,b99) by Def6;
    then reconsider c99= (Padd(o)).((f.a),(f.b)) as Element of AV( ADG);
    reconsider c9=c99 as Element of ADG by A3;
    o,a99 // b99,c99 by A4,Def5;
    then [[o9,a9],[b9,c9]] in CONGRD(ADG) by A2,A3,ANALOAF:def 2;
    then
A5: o9+c9 = a9+b9 by TDGROUP:def 2;
    a9 = o9+a & b9 = o9+b by A1;
    then o9+c9 = (o9+((a+o9)+b)) by A5,RLVECT_1:def 3
      .= o9+(o9+(a+b)) by RLVECT_1:def 3;
    then c9 = o9+(a+b) by RLVECT_1:8
      .= f.(a+b) by A1;
    hence thesis;
  end;
  f.(0.ADG) = o9+(0.ADG) by A1
    .= 0.(GroupVect(AV(ADG),o)) by A2,RLVECT_1:4;
  hence f.(0.ADG) = 0.(GroupVect(AV(ADG),o));
  thus f.(-a) = (Pcom(o)).(f.a)
  proof
A6: ((Pcom(o)).(f.a)) = Pcom(o,a99) by Def7;
    then reconsider c99 = (Pcom(o)).(f.a) as Element of AV(ADG);
    reconsider c9=c99 as Element of ADG by A3;
    a99,o // o,c99 by A6,Lm1;
    then [[a9,o9],[o9,c9]] in CONGRD(ADG) by A2,A3,ANALOAF:def 2;
    then a9+c9 = o9+o9 by TDGROUP:def 2;
    then
A7: o9+o9 = (o9+a)+c9 by A1
      .= o9+(a+c9) by RLVECT_1:def 3;
    f.(-a) = o9+(-a) by A1
      .= (c9+a)+(-a) by A7,RLVECT_1:8
      .= c9+(a+(-a)) by RLVECT_1:def 3
      .= c9+(0.ADG) by RLVECT_1:5
      .= c9 by RLVECT_1:4;
    hence thesis;
  end;
end;
