theorem Th121:
  H + a = H + b iff H + a meets H + b
proof
  thus H + a = H + b implies H + a meets H + b by Th108;
  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,Th12
    .= -g + h + b by RLVECT_1:def 3;
  then a + -b = -g + h by Th13;
  hence thesis by A5,A7,Th50,Th120;
end;
