reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th04:
  for X be set, B be non empty Subset-Family of X,
  L be Subset of BoolePoset X st B=L holds <.B.]=uparrow L
  proof
    let X be set,B be non empty Subset-Family of X,
    L be Subset of BoolePoset X;
    assume
A1: B=L;
    hereby let x be object;
      assume
A2:   x in <.B.];
      then reconsider x0=x as Subset of X;
      consider b be Element of B such that
A3:   b c= x0 by A2,def3;
      reconsider b1=b as Element of BoolePoset X
      by LATTICE3:def 1;
      reconsider x1=x0 as Element of BoolePoset X by LATTICE3:def 1;
      b1 <= x1 & b1 in L by A3,A1,YELLOW_1:2;
      hence x in uparrow L by WAYBEL_0:def 16;
    end;
    let x be object;
    assume
A4B1: x in uparrow L;
    then reconsider x0=x as Element of BoolePoset X;
    consider y be Element of BoolePoset X such that
A5B2: y <= x0 and
A6B3: y in L by A4B1,WAYBEL_0:def 16;
A7B4: y c= x0 by A5B2,YELLOW_1:2;
    x0 is Element of bool X by LATTICE3:def 1;
    then reconsider x1=x as Subset of X;
    x1 in <.B.] by A1,A6B3,A7B4,def3;
    hence x in <.B.];
  end;
