theorem
  (gO in the_LeftOptions_of -g iff -gO in the_RightOptions_of g)
    & (gO in the_LeftOptions_of g iff -gO in the_RightOptions_of -g)
    & (gO in the_RightOptions_of -g iff -gO in the_LeftOptions_of g)
    & (gO in the_RightOptions_of g iff -gO in the_LeftOptions_of -g)
proof
  g = -(-g) & gO = -(-gO) by Th40;
  hence thesis by Th39;
end;
