theorem Th115:
  a * H = b * H iff a * H meets b * H
proof
  a * H <> {} by Th108;
  hence a * H = b * H implies a * H meets b * H;
  assume a * H meets b * H;
  then consider x being object such that
A1: x in a * H and
A2: x in b * H by XBOOLE_0:3;
  reconsider x as Element of G by A2;
  consider g such that
A3: x = a * g and
A4: g in H by A1,Th103;
A5: g" in H by A4,Th51;
  consider h being Element of G such that
A6: x = b * h and
A7: h in H by A2,Th103;
  a = b * h * g" by A3,A6,GROUP_1:14
    .= b * (h * g") by GROUP_1:def 3;
  then b" * a = h * g" by GROUP_1:13;
  then b" * a in H by A7,A5,Th50;
  hence thesis by Th114;
end;
