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;
reserve x,y for Element of vectgroup(M);
reserve w for Function of [:S,S:],the carrier of G;
reserve a,b,c for Point of MidStr(#S,@(w)#);
reserve M for non empty MidStr;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve a,b,b1,b2,c for Point of M;
reserve M for MidSp;
reserve W for ATLAS of M;
reserve a,b,b1,b2,c,d for Point of M;
reserve x for Vector of W;

theorem
  a@c = b1@b2 iff W.(a,c) = W.(a,b1) + W.(a,b2)
proof
  set w = the function of W, G = the algebra of W;
A1: G is midpoint_operator add-associative right_zeroed right_complementable
  Abelian by Def12;
  w is_atlas_of the carrier of M,G & w is associating by Def12;
  hence thesis by A1,Th27;
end;
