theorem
  a@c = b iff W.(a,c) = Double W.(a,b)
proof
  set w = the function of W, G = the algebra of W;
A1: G is midpoint_operator add-associative right_zeroed right_complementable
  Abelian by Def12;
  w is_atlas_of the carrier of M,G & w is associating by Def12;
  hence thesis by A1,Th28;
end;
