theorem Th8:
  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 p@q = q@p
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;
  set r = p@q;
  w.(p,r) = w.(r,q) by A2;
  then w.(r,p) = w.(q,r) by A1,Th5;
  hence thesis by A2;
end;
