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;
