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 Th53:
  for o9 being Element of ADG st (for b being Element of ADG holds
  f.b = o9+b) holds f is one-to-one
proof
  let o9 be Element of ADG such that
A1: for b being Element of ADG holds f.b = o9+b;
  now
    let x1,x2 be object such that
A2: x1 in dom(f) & x2 in dom(f) and
A3: f.x1 = f.x2;
    reconsider x19=x1,x29=x2 as Element of ADG by A2,FUNCT_2:def 1;
    o9+x29 = f.x19 by A1,A3
      .= o9+x19 by A1;
    hence x1=x2 by RLVECT_1:8;
  end;
  hence thesis by FUNCT_1:def 4;
end;
