reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th15: ::3.9 (1-2), p.70
  for L be up-complete lower-bounded LATTICE, X be Subset of L
holds X is order-generating iff for l being Element of L ex Y be Subset of X st
  l = "/\" (Y,L)
proof
  let L be up-complete lower-bounded LATTICE, X be Subset of L;
  thus X is order-generating implies for l being Element of L ex Y be Subset
  of X st l = "/\" (Y,L)
  proof
    assume
A1: X is order-generating;
    let l be Element of L;
    for x being object st x in ((uparrow l) /\ X)
     holds x in X by XBOOLE_0:def 4;
    then reconsider Y = ((uparrow l) /\ X) as Subset of X by TARSKI:def 3;
    l = "/\" (Y,L) by A1;
    hence thesis;
  end;
  thus (for l being Element of L ex Y be Subset of X st l = "/\" (Y,L))
  implies X is order-generating
  proof
    assume
A2: for l being Element of L ex Y be Subset of X st l = "/\" (Y,L);
    let l be Element of L;
    consider Y be Subset of X such that
A3: l = "/\" (Y,L) by A2;
    set S = ((uparrow l) /\ X);
    thus ex_inf_of S,L by YELLOW_0:17;
A4: for b be Element of L st b is_<=_than S holds b <= l
    proof
      let b be Element of L;
      assume
A5:   b is_<=_than S;
      now
        let x be Element of L;
        assume
A6:     x in Y;
        l is_<=_than Y by A3,YELLOW_0:33;
        then l <= x by A6;
        then x in uparrow l by WAYBEL_0:18;
        then x in S by A6,XBOOLE_0:def 4;
        hence b <= x by A5;
      end;
      then b is_<=_than Y;
      hence thesis by A3,YELLOW_0:33;
    end;
    now
      let x be Element of L;
      assume x in S;
      then x in (uparrow l) by XBOOLE_0:def 4;
      hence l <= x by WAYBEL_0:18;
    end;
    then l is_<=_than S;
    hence thesis by A4,YELLOW_0:33;
  end;
end;
