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 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;
