theorem Th4:
  w is_atlas_of S,G implies w.(a,b) = -w.(b,a)
proof
  assume
A1: w is_atlas_of S,G;
  then w.(b,a) + w.(a,b) = w.(b,b)
    .= 0.G by A1,Th2;
  then -w.(b,a) = --w.(a,b) by RLVECT_1:6;
  hence thesis by RLVECT_1:17;
end;
