theorem Th28:
  w is_atlas_of the carrier of M,G & w is associating
  implies (a@c = b iff w.(a,c) = Double w.(a,b))
proof
  assume
A1: w is_atlas_of the carrier of M,G & w is associating;
  then reconsider MM = M as MidSp by Th20;
  reconsider bb = b as Point of MM;
  bb@bb = bb by MIDSP_1:def 3;
  hence thesis by A1,Th27;
end;
