theorem Th3:
  w is_atlas_of S,G & w.(a,b) = 0.G implies a = b
proof
  assume that
A1: w is_atlas_of S,G and
A2: w.(a,b) = 0.G;
  w.(a,b) = w.(a,a) by A1,A2,Th2;
  hence thesis by A1;
end;
