theorem Th115:
  a + H = b + H iff a + H meets b + H
proof
  thus a + H = b + H implies a + H meets b + H by Th108;
  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,Th13
    .= b + (h + -g) by RLVECT_1:def 3;
  then -b + a = h + -g by Th12;
  hence thesis by A5,A7,Th50,Th114;
end;
