theorem Th25:
  w is_atlas_of S,G implies Atlas(w) is associating
proof
  assume A1: w is_atlas_of S,G;
  for a,b,c holds a@b = c iff (Atlas(w)).(a,c) = (Atlas(w)).(c,b)
  proof
    let a,b,c;
    @(w).(a,b) = c iff w.(a,c) = w.(c,b) by A1,Th24;
    hence thesis by MIDSP_1:def 1;
  end;
  hence thesis;
end;
