theorem Th18:
  g + h = h + g iff -g + -h = -h + -g
proof
  thus g + h = h + g implies -g + -h = -h + -g
  proof
    assume
A1: g + h = h + g;
    hence -g + -h = -(g + h) by Th16
      .= -h + -g by A1,Th17;
  end;
  assume
A2: -g + -h = -h + -g;
  thus g + h = - -(g + h) .= -(-h + -g) by Th16
    .= - -h + - -g by A2,Th16
    .= h + g;
end;
