theorem Th133:
  for H being finite Subgroup of G ex B,C being finite set st B =
  a + H & C = H + a & card H = card B & card H = card C
proof
  let H be finite Subgroup of G;
  reconsider B = a + H, C = H + a as finite set by Th131,CARD_1:38;
  take B,C;
  thus thesis by Th131,CARD_1:5;
end;
