reserve X for set;

theorem
  for Y be Subset-Family of X holds InclPoset Y is full SubRelStr of
  BoolePoset X
proof
  set L = BoolePoset X;
  let Y be Subset-Family of X;
  reconsider Y9 = Y as Subset of L by LATTICE3:def 1;
  the carrier of L = bool X by LATTICE3:def 1;
  then reconsider In = the InternalRel of L as Relation of bool X;
  for x be object st x in bool X ex y be object st [x,y] in In
  proof
    let x be object;
    assume x in bool X;
    then reconsider x9 = x as Element of L by LATTICE3:def 1;
    take y = x9;
    x9 <= y;
    hence thesis by ORDERS_2:def 5;
  end;
  then
A1: dom In = bool X by RELSET_1:9;
A2: now
    let Y,Z be set;
    assume Y in bool X & Z in bool X;
    then reconsider Y9 = Y, Z9 = Z as Element of L by LATTICE3:def 1;
    thus [Y,Z] in the InternalRel of L implies Y c= Z
    proof
      assume [Y,Z] in the InternalRel of L;
      then Y9 <= Z9 by ORDERS_2:def 5;
      hence thesis by Th2;
    end;
    thus Y c= Z implies [Y,Z] in the InternalRel of L
    proof
      assume Y c= Z;
      then Y9 <= Z9 by Th2;
      hence thesis by ORDERS_2:def 5;
    end;
  end;
  for y be object st y in bool X ex x be object st [x,y] in In
  proof
    let y be object;
    assume y in bool X;
    then reconsider y9 = y as Element of L by LATTICE3:def 1;
    take x = y9;
    x <= y9;
    hence thesis by ORDERS_2:def 5;
  end;
  then field the InternalRel of L = bool X \/ bool X by A1,RELSET_1:10;
  then
A3: the InternalRel of L = RelIncl bool X by A2,WELLORD2:def 1;
  RelStr(#Y9,(the InternalRel of L) |_2 Y9#) is full SubRelStr of L by
YELLOW_0:56;
  hence thesis by A3,WELLORD2:7;
end;
