
theorem Th4:
  for X being set, B being non empty Subset-Family of X st B is
cap-closed for x being Element of B st x is_/\-irreducible_in B & x <> X for S
  being finite Subset-Family of X st S c= B & x = Intersect S holds x in S
proof
  let X be set, B be non empty Subset-Family of X such that
A1: B is (B2);
  let x be Element of B such that
A2: x is_/\-irreducible_in B and
A3: x <> X;
  defpred P[set] means (ex a, b being Element of B st x <> a & x <> b & x = a
  /\ b) or ex f being Subset-Family of X st $1 = {} or $1 <> {} & $1 = f &
  Intersect f <> x & Intersect f in B;
  let S be finite Subset-Family of X such that
A4: S c= B and
A5: x = Intersect S and
A6: not x in S;
A7: now
    let s, A be set such that
A8: s in S and
    A c= S and
A9: P[A];
    per cases by A9;
    suppose
      ex a, b being Element of B st x <> a & x <> b & x = a /\ b;
      hence P[A\/{s}];
    end;
    suppose
      ex f being Subset-Family of X st A = {} or A = f & Intersect f
      <> x & Intersect f in B;
      then consider f being Subset-Family of X such that
A10:  A = {} or A <> {} & A = f & Intersect f <> x & Intersect f in B;
      thus P[A\/{s}]
      proof
        reconsider sf = {s} as Subset-Family of X by A8,ZFMISC_1:31;
A11:    Intersect sf = meet sf by SETFAM_1:def 9;
        then
A12:    Intersect sf = s by SETFAM_1:10;
        per cases by A10;
        suppose
          A = {};
          hence thesis by A4,A6,A8,A12;
        end;
        suppose
A13:      A <> {} & A = f & Intersect f <> x & Intersect f in B;
          then reconsider As = A\/sf as non empty Subset-Family of X by
XBOOLE_1:8;
A14:      Intersect As = meet As by SETFAM_1:def 9
            .= meet A /\ meet sf by A13,SETFAM_1:9;
A15:      Intersect f = meet f by A13,SETFAM_1:def 9;
          thus P[A\/{s}]
          proof
            per cases;
            suppose
A16:          Intersect As <> x;
              Intersect As in B by A1,A4,A8,A11,A12,A13,A15,A14;
              hence thesis by A16;
            end;
            suppose
              Intersect As = x;
              hence thesis by A4,A6,A8,A11,A12,A13,A15,A14;
            end;
          end;
        end;
      end;
    end;
  end;
A17: P[{}];
A18: S is finite;
  P[S] from FINSET_1:sch 2(A18,A17,A7);
  hence thesis by A2,A3,A5,SETFAM_1:def 9;
end;
