reserve G for non empty addLoopStr;
reserve x for Element of G;
reserve M for non empty MidStr;
reserve p,q,r for Point of M;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve S for non empty set;
reserve a,b,b9,c,c9,d for Element of S;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable Abelian non
  empty addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;
reserve M for MidSp;
reserve p,q,r,s for Point of M;
reserve G for midpoint_operator add-associative right_zeroed
  right_complementable Abelian non empty addLoopStr;
reserve x,y for Element of G;

theorem Th16:
  for G being Fanoian add-associative right_zeroed
right_complementable non empty addLoopStr, x being Element of G holds x = -x
  implies x = 0.G
proof
  let G be Fanoian add-associative right_zeroed right_complementable non
  empty addLoopStr, x be Element of G;
A1: -x + x = 0.G by RLVECT_1:5;
  assume x = -x;
  hence thesis by A1,VECTSP_1:def 18;
end;
