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;

theorem Th9:
  for w being Function of [:the carrier of M,the carrier of M:],
  the carrier of G holds w is_atlas_of the carrier of M,G &
  w is associating implies ex r st r@p = q
proof
  let w be Function of [:the carrier of M,the carrier of M:],the carrier of G;
  assume that
A1: w is_atlas_of the carrier of M,G and
A2: w is associating;
  consider r such that
A3: w.(r,q) = w.(q,p) by A1,Th6;
  take r;
  thus thesis by A2,A3;
end;
