theorem Th5:
  w is_atlas_of S,G & w.(a,b) = w.(c,d) implies w.(b,a) = w.(d,c)
proof
  assume that
A1: w is_atlas_of S,G and
A2: w.(a,b) = w.(c,d);
  thus w.(b,a) = -w.(c,d) by A1,A2,Th4
    .= w.(d,c) by A1,Th4;
end;
