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;
  carr(H),a * H are_equipotent & carr(H),H * a are_equipotent by Th131;
  hence thesis by CARD_1:5;
end;
