reserve a, b, r, s for Real;
reserve n, m for Nat,
  F for Subset-Family of Closed-Interval-TSpace (r,s);

theorem Th47:
  for L being TopSpace, G, G1 being Subset-Family of L st G is
Cover of L & G is finite for ALL being set st G1 = G \ {X where X is Subset of
L: X in G & ex Y being Subset of L st Y in G & X c< Y} & ALL = {C where C is
  Subset-Family of L: C is Cover of L & C c= G1} holds ALL
  has_lower_Zorn_property_wrt RelIncl ALL
proof
  let L be TopSpace;
  let G, G1 be Subset-Family of L;
  assume that
A1: G is Cover of L and
A2: G is finite;
  let ALL be set;
  set ZAW = {X where X is Subset of L: X in G & ex Y being Subset of L st Y in
  G & X c< Y};
  assume that
A3: G1 = G \ ZAW and
A4: ALL = {C where C is Subset-Family of L: C is Cover of L & C c= G1};
A5: G1 is Cover of L by A1,A2,A3,Th45;
  set R = RelIncl ALL;
A6: field R = ALL by WELLORD2:def 1;
  let Y be set such that
A7: Y c= ALL and
A8: R |_2 Y is being_linear-order;
  per cases;
  suppose
A9: Y is non empty;
    defpred A[set] means $1 is non empty implies meet $1 in Y;
    set E = {F(D) where D is Subset-Family of L: D in bool G1};
    take x = meet Y;
A10: ALL c= E
    proof
      let a be object;
      assume a in ALL;
      then ex C being Subset-Family of L st a = C & C is Cover of L & C c= G1
      by A4;
      hence thesis;
    end;
A11: bool G1 is finite by A2,A3;
    E is finite from FRAENKEL:sch 21(A11);
    then
A12: Y is finite by A7,A10;
A13: for x, B being set st x in Y & B c= Y & A[B] holds A[B \/ {x}]
    proof
      let x, B be set such that
A14:  x in Y and
      B c= Y and
A15:  A[B] and
      B \/ {x} is non empty;
      per cases;
      suppose
        B is empty;
        hence thesis by A14,SETFAM_1:10;
      end;
      suppose
A16:    B is non empty;
        R |_2 Y is connected by A8;
        then
A17:    R |_2 Y is_connected_in field (R |_2 Y);
        field (R |_2 Y) = Y by A6,A7,ORDERS_1:71;
        then [x,meet B] in R |_2 Y or [meet B,x] in R |_2 Y or x = meet B by
A14,A15,A16,A17;
        then [x,meet B] in R or [meet B,x] in R or x = meet B by XBOOLE_0:def 4
;
        then
A18:    meet B c= x or x c= meet B by A7,A14,A15,A16,WELLORD2:def 1;
        meet (B \/ {x}) = meet B /\ meet {x} by A16,SETFAM_1:9
          .= meet B /\ x by SETFAM_1:10;
        hence thesis by A14,A15,A16,A18,XBOOLE_1:28;
      end;
    end;
    consider y being object such that
A19: y in Y by A9;
    y in ALL by A7,A19;
    then
A20: ex C being Subset-Family of L st y = C & C is Cover of L & C c= G1 by A4;
    then reconsider X = x as Subset-Family of L by A19,SETFAM_1:7;
A21: A[{}];
A22: A[Y] from FINSET_1:sch 2(A12,A21,A13);
A23: X is Cover of L
    proof
      let a be object;
      assume
A24:  a in the carrier of L;
      meet Y in ALL by A7,A9,A22;
      then consider C being Subset-Family of L such that
A25:  meet Y = C and
A26:  C is Cover of L and
      C c= G1 by A4;
      the carrier of L c= union C by A26,SETFAM_1:def 11;
      hence thesis by A24,A25;
    end;
    x c= G1 by A19,A20,SETFAM_1:7;
    hence
A27: x in ALL by A4,A23;
    let y be set;
    assume
A28: y in Y;
    then x c= y by SETFAM_1:7;
    hence thesis by A7,A27,A28,WELLORD2:def 1;
  end;
  suppose
A29: Y is empty;
    take G1;
    thus thesis by A4,A5,A29;
  end;
end;
