theorem ThX7:
  A <> {} iff -A <> {}
proof
  set x = the Element of -A;
  thus A <> {} implies -A <> {}
  proof
    set x = the Element of A;
    assume
A1: A <> {};
    then reconsider x as Element of G by Lm1;
    -x in -A by A1;
    hence thesis;
  end;
  assume -A <> {};
  then ex a st x = -a & a in A by Th2;
  hence thesis;
end;
