reserve X for non empty set;

theorem Th2:
  for B being non empty Subset-Family of X st
  (for B1,B2 being Element of B holds
    ex BB being Subset of B st B1/\B2 = union BB) & X = union B
  holds FinMeetCl B c= UniCl B
  proof
    let B be non empty Subset-Family of X;
    assume that
A1: for B1,B2 be Element of B
    ex BB being Subset of B st B1/\B2=union BB and
A2: X = union B;
    let x be object;
    assume
A3: x in FinMeetCl B;
    then reconsider x0=x as Subset of X;
    consider Y be Subset-Family of X such that
A4: Y c= B and
A5: Y is finite and
A6: x0 = Intersect Y by A3,CANTOR_1:def 3;
    defpred PP[Nat] means
    for Y be Subset-Family of X,x be Subset of X st
    Y c= B & card Y=$1 & x = Intersect Y holds
    x in UniCl B;
A7: PP[0]
    proof
      let Y be Subset-Family of X,x be Subset of X;
      assume that
      Y c= B and
A8:   card Y=0 and
A9:   x = Intersect Y;
      Y={} by A8; then
A10:  x = X by A9,SETFAM_1:def 9;
      reconsider x0=x as set;
      thus x in UniCl B by A2,A10,CANTOR_1:def 1;
    end;
A11:
    for k being Nat holds PP[k] implies PP[k+1]
    proof
      let k be Nat;
      assume
A12:  PP[k];
      let Y be Subset-Family of X,x be Subset of X;
      assume that
A13:  Y c= B and
A14:  card Y=k+1 and
A15:  x = Intersect Y;
      Y is finite set by A14;
      then consider x1 be Element of Y,C being Subset of Y such that
A16:  Y=C\/{x1} and
A17:  card C = k by A14,PRE_CIRC:24;
A18:  C c= B & card C=k by A13,A17;
      B c= bool X;
      then C c= bool X by A13;
      then reconsider C0=C as Subset-Family of X;
      per cases;
      suppose
A19:    C = {};
        meet {x1}=x1 by SETFAM_1:10; then
A20:    Intersect Y = x1 by A19,A16,SETFAM_1:def 9;
        then x1 in bool X;
        then {x1} c= bool X by TARSKI:def 1;
        then reconsider B0={x1} as Subset-Family of X;
        x1 in Y by A16; then
A21:    {x1} c= Y & Y c= B by A13,TARSKI:def 1;
        x in B & B0 c= B & x=union B0 by A16,A21,A15,A20;
        hence x in UniCl B by CANTOR_1:def 1;
      end;
      suppose
A22:    C<>{};
        then meet(C\/{x1})=meet(C) /\ meet({x1}) by SETFAM_1:9; then
A23:    meet(Y)=meet C /\ x1 by A16,SETFAM_1:10;
        meet Y =Intersect(Y) by A16,SETFAM_1:def 9; then
A24:    Intersect(Y)=Intersect(C0)/\x1 by A22,A23,SETFAM_1:def 9;
        Intersect(Y) in UniCl B
        proof
          Intersect(C0) in UniCl B by A18,A12;
          then consider Y2 be Subset-Family of X such that
A25:      Y2 c= B and
A26:      Intersect(C0)=union Y2 by CANTOR_1:def 1;
          per cases;
          suppose
A27:        Y2 is empty;
            {} c= X;
            then reconsider x0={} as Subset of X;
A28:        {} c= bool X & {} c= B & {}=union {} by ZFMISC_1:2;
            then reconsider S={} as Subset-Family of X;
            thus thesis by A27,A24,A26,A28,CANTOR_1:def 1;
          end;
          suppose
A29:        Y2 is non empty;
            set Y3=the set of all y/\x1 where y is Element of Y2;
            Y3 c= bool X
            proof
              let x be object;
              assume
A30:          x in Y3;
              then reconsider x as Element of Y3;
              consider y be Element of Y2 such that
A31:          x=y/\x1 by A30;
A32:          x c= x1 by A31,XBOOLE_1:17;
              x1 in Y & Y c= bool X by A16;
              then x1 c= X;
              then x c= X by A32;
              hence thesis;
            end;
            then reconsider Y3 as Subset-Family of X;
A33:        union Y3=(union Y2) /\ x1
            proof
              hereby let x be object;
                assume
A34:            x in union Y3;
                consider a1 be set such that
A35:            x in a1 and
A36:            a1 in Y3 by A34,TARSKI:def 4;
                consider x2 be Element of Y2 such that
A37:            a1=x2/\x1 by A36;
                x in a1 & a1 c= x2 & a1 c= x1 by A35,A37,XBOOLE_1:17;
                then x in union Y2 & x in x1 by A29,TARSKI:def 4;
                hence x in (union Y2)/\x1 by XBOOLE_0:def 4;
              end;
              let x be object;
              assume x in (union Y2)/\ x1; then
A38:          x in union Y2 & x in x1 by XBOOLE_0:def 4;
              then consider a be set such that
A39:          x in a and
A40:          a in Y2 by TARSKI:def 4;
              reconsider a as Element of Y2 by A40;
A41:          x in a/\x1 by A38,A39,XBOOLE_0:def 4;
              a/\x1 in Y3;
              hence x in union Y3 by A41,TARSKI:def 4;
            end;
A42:        Intersect(Y)=union Y3
            proof
              hereby let t be object;
                assume t in Intersect(Y); then
A43:            t in union Y2 & t in x1 by A24,A26,XBOOLE_0:def 4;
                then consider t0 be set such that
A44:            t in t0 and
A45:            t0 in Y2 by TARSKI:def 4;
A46:            t in t0/\x1 by A43,A44,XBOOLE_0:def 4;
                t0/\x1 in Y3 by A45;
                hence t in union Y3 by A46,TARSKI:def 4;
              end;
              let t be object;
              assume t in union Y3;
              then t in union Y2 & t in x1 by A33,XBOOLE_0:def 4;
              then t in meet C0 & t in x1 by A26,A22,SETFAM_1:def 9;
              then t in meet(C0) /\ x1 by XBOOLE_0:def 4;
              hence t in Intersect Y by A23,A16,SETFAM_1:def 9;
            end;
            Y3 c= UniCl B
            proof
              let t be object;
              assume t in Y3;
              then consider a be Element of Y2 such that
A47:          t=a/\x1;
              reconsider a2=a as Element of B by A29,A25;
              reconsider x2=x1 as Element of B by A13,A16;
              consider BP being Subset of B such that
A48:          a2/\x2 = union BP by A1;
              reconsider ax=a2/\x2 as Subset of X;
              BP c= B & B c= bool X; then
A49:          BP c= bool X;
              thus t in UniCl B by A47,A48,A49,CANTOR_1:def 1;
            end;
            then Intersect(Y) in UniCl UniCl B by A42,CANTOR_1:def 1;
            hence thesis by YELLOW_9:15;
          end;
        end;
        hence x in UniCl B by A15;
      end;
    end;
A50: for k being Nat holds PP[k] from NAT_1:sch 2(A7,A11);
    reconsider CY=card Y as Nat by A5;
    PP[CY] by A50;
    hence x in UniCl B by A4,A6;
  end;
