theorem Th27:
  w is_atlas_of the carrier of M,G & w is associating
  implies (a@c = b1@b2 iff w.(a,c) = w.(a,b1) + w.(a,b2))
proof
  assume that
A1: w is_atlas_of the carrier of M,G and
A2: w is associating;
A3: a@c = b1@b2 iff w.(a,b2) = w.(b1,c) by A1,A2,Th13;
  hence a@c = b1@b2 implies w.(a,c) = w.(a,b1) + w.(a,b2) by A1;
  w.(a,c) = w.(a,b1) + w.(b1,c) by A1;
  hence thesis by A3,RLVECT_1:8;
end;
