reserve G for non empty addLoopStr;
reserve x for Element of G;
reserve M for non empty MidStr;
reserve p,q,r for Point of M;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve S for non empty set;
reserve a,b,b9,c,c9,d for Element of S;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;

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;
