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

theorem ::3.9 (1-3), p.70
  for L be up-complete lower-bounded LATTICE, X be Subset of L holds X
  is order-generating iff for Y be Subset of L st X c= Y & for Z be Subset of Y
  holds "/\" (Z,L) in Y holds the carrier of L = Y
proof
  let L be up-complete lower-bounded LATTICE, X be Subset of L;
  thus X is order-generating implies for Y be Subset of L st X c= Y & for Z be
  Subset of Y holds "/\"(Z,L) in Y holds the carrier of L = Y
  proof
    assume
A1: X is order-generating;
    let Y be Subset of L;
    assume that
A2: X c= Y and
A3: for Z be Subset of Y holds "/\"(Z,L) in Y;
    now
      let l1 be object;
      assume l1 in the carrier of L;
      then reconsider l = l1 as Element of L;
      (uparrow l) /\ Y c= Y & (uparrow l) /\ X c= (uparrow l) /\ Y by A2,
XBOOLE_1:17,26;
      then
A4:   (uparrow l) /\ X c= Y;
      l = inf ((uparrow l) /\ X) by A1;
      hence l1 in Y by A3,A4;
    end;
    hence the carrier of L c= Y;
    thus thesis;
  end;
  thus (for Y be Subset of L st X c= Y & for Z be Subset of Y holds "/\"(Z,L)
  in Y holds the carrier of L = Y) implies X is order-generating
  proof
    set Y = {"/\"(Z,L) where Z is Subset of L : Z c= X};
    now
      let x be object;
      assume x in Y;
      then ex Z be Subset of L st x = "/\"(Z,L) & Z c= X;
      hence x in the carrier of L;
    end;
    then reconsider Y as Subset of L by TARSKI:def 3;
    now
      let x be object;
      assume
A5:   x in X;
      then reconsider x1 = x as Element of L;
      reconsider x2 = {x1} as Subset of L;
A6:   x1 = "/\"(x2,L) by YELLOW_0:39;
      {x1} c= X by A5,ZFMISC_1:31;
      hence x in Y by A6;
    end;
    then
A7: X c= Y;
    assume
A8: for Y be Subset of L st X c= Y & for Z be Subset of Y holds "/\"(Z
    ,L) in Y holds the carrier of L = Y;
    for l being Element of L ex Z be Subset of X st l = "/\" (Z,L)
    proof
      let l be Element of L;
      for Z be Subset of Y holds "/\"(Z,L) in Y
      proof
        let Z be Subset of Y;
        set S = union {A where A is Subset of L : A c= X & "/\"(A,L) in Z};
        now
          let x be object;
          assume x in S;
          then consider Y be set such that
A9:       x in Y and
A10:      Y in {A where A is Subset of L : A c= X & "/\" (A,L) in Z}
          by TARSKI:def 4;
          ex A be Subset of L st Y = A & A c= X & "/\"(A,L) in Z by A10;
          hence x in the carrier of L by A9;
        end;
        then reconsider S as Subset of L by TARSKI:def 3;
        defpred P[Subset of L] means $1 c= X & "/\"($1,L) in Z;
        set N = {"/\"(A,L) where A is Subset of L : P[A]};
        now
          let x be object;
          assume
A11:      x in Z;
          then x in Y;
          then ex Z be Subset of L st x = "/\"(Z,L) & Z c= X;
          hence x in N by A11;
        end;
        then
A12:    Z c= N;
        now
          let B be set;
          assume B in {A where A is Subset of L : A c= X & "/\"(A,L) in Z};
          then ex A be Subset of L st B = A & A c= X & "/\"(A, L) in Z;
          hence B c= X;
        end;
        then
A13:    S c= X by ZFMISC_1:76;
        now
          let x be object;
          assume x in N;
          then ex S be Subset of L st x = "/\"(S,L) & S c= X & "/\"(S, L) in Z;
          hence x in Z;
        end;
        then N c= Z;
        then "/\"(Z,L) = "/\"(N,L) by A12,XBOOLE_0:def 10
          .= "/\" (union {A where A is Subset of L : P[A]}, L) from YELLOW_3
        :sch 3;
        hence thesis by A13;
      end;
      then the carrier of L = Y by A8,A7;
      then l in Y;
      then ex Z be Subset of L st l = "/\" (Z,L) & Z c= X;
      hence thesis;
    end;
    hence thesis by Th15;
  end;
end;
