theorem Th16:
  {}(the carrier of G) * A = {} & A * {}(the carrier of G) = {}
proof
A1: now
    set x = the Element of A * {}(the carrier of G);
    assume A * {}(the carrier of G) <> {};
    then ex g1,g2 st x = g1 * g2 & g1 in A & g2 in {}(the carrier of G) by Th8;
    hence contradiction;
  end;
  now
    set x = the Element of {}(the carrier of G) * A;
    assume {}(the carrier of G) * A <> {};
    then ex g1,g2 st x = g1 * g2 & g1 in {}(the carrier of G) & g2 in A by Th8;
    hence contradiction;
  end;
  hence thesis by A1;
end;
