theorem
  H * a = H * b iff H * a meets H * b
proof
  H * a <> {} by Th108;
  hence H * a = H * b implies H * a meets H * b;
  assume H * a meets H * b;
  then consider x being object such that
A1: x in H * a and
A2: x in H * b by XBOOLE_0:3;
  reconsider x as Element of G by A2;
  consider g such that
A3: x = g * a and
A4: g in H by A1,Th104;
A5: g" in H by A4,Th51;
  consider h being Element of G such that
A6: x = h * b and
A7: h in H by A2,Th104;
  a = g" * (h * b) by A3,A6,GROUP_1:13
    .= g" * h * b by GROUP_1:def 3;
  then a * b" = g" * h by GROUP_1:14;
  then a * b" in H by A7,A5,Th50;
  hence thesis by Th120;
end;
