
theorem Th29:
  for S being meet-continuous Scott TopLattice, F being finite
Subset of S holds Int uparrow F c= union { wayabove x where x is Element of S :
  x in F }
proof
  let S be meet-continuous Scott TopLattice, F be finite Subset of S;
  defpred P[set] means ex UU being Subset-Family of S st UU = {uparrow x where
x is Element of S : x in $1} & (union UU)^0 = union { (uparrow x)^0 where x is
  Element of S : x in $1 };
A1: for b, J being set st b in F & J c= F & P[J] holds P[J \/ {b}]
  proof
    let b, J be set;
    assume that
A2: b in F and
    J c= F;
    reconsider bb = b as Element of S by A2;
A3: (union {(uparrow x)^0 where x is Element of S : x in J}) \/ ((uparrow
    bb)^0) = union {(uparrow x)^0 where x is Element of S : x in J \/ {b}}
    proof
      {(uparrow x)^0 where x is Element of S : x in J} c= {(uparrow x)^0
      where x is Element of S : x in J \/ {b}}
      proof
        let a be object;
        assume a in {(uparrow x)^0 where x is Element of S : x in J};
        then
A4:     ex y being Element of S st a = (uparrow y)^0 & y in J;
        J c= J \/ {b} by XBOOLE_1:7;
        hence thesis by A4;
      end;
      then
A5:   union {(uparrow x)^0 where x is Element of S : x in J} c= union {(
      uparrow x)^0 where x is Element of S : x in J \/ {b}} by ZFMISC_1:77;
A6:   {b} c= J \/ {b} by XBOOLE_1:7;
      b in {b} by TARSKI:def 1;
      then (uparrow bb)^0 in {(uparrow x)^0 where x is Element of S : x in J
      \/ {b}} by A6;
      then (uparrow bb)^0 c= union {(uparrow x)^0 where x is Element of S : x
      in J \/ {b}} by ZFMISC_1:74;
      hence (union {(uparrow x)^0 where x is Element of S : x in J}) \/ ((
uparrow bb)^0) c= union {(uparrow x)^0 where x is Element of S : x in J \/ {b}}
      by A5,XBOOLE_1:8;
      let a be object;
      assume
      a in union {(uparrow x)^0 where x is Element of S : x in J \/ { b}};
      then consider A being set such that
A7:   a in A and
A8:   A in {(uparrow x)^0 where x is Element of S : x in J \/ {b}} by
TARSKI:def 4;
      consider y being Element of S such that
A9:   A = (uparrow y)^0 and
A10:  y in J \/ {b} by A8;
      per cases by A10,XBOOLE_0:def 3;
      suppose
        y in J;
        then
        (uparrow y)^0 in {(uparrow x)^0 where x is Element of S : x in J};
        then
A11:    a in union {(uparrow x)^0 where x is Element of S : x in J } by A7,A9,
TARSKI:def 4;
        union {(uparrow x)^0 where x is Element of S : x in J} c= (union
{(uparrow x)^0 where x is Element of S : x in J}) \/ ((uparrow bb)^0) by
XBOOLE_1:7;
        hence thesis by A11;
      end;
      suppose
A12:    y in {b};
A13:    (uparrow bb)^0 c= (union {(uparrow x)^0 where x is Element of S :
        x in J}) \/ ((uparrow bb)^0) by XBOOLE_1:7;
        y = b by A12,TARSKI:def 1;
        hence thesis by A13,A7,A9;
      end;
    end;
    assume P[J];
    then consider UU being Subset-Family of S such that
A14: UU = {uparrow x where x is Element of S : x in J} and
A15: (union UU)^0 = union {(uparrow x)^0 where x is Element of S : x in J};
    reconsider I = UU \/ {uparrow bb} as Subset-Family of S;
    take I;
    thus I = {uparrow x where x is Element of S : x in J \/ {b}}
    proof
      thus I c= {uparrow x where x is Element of S : x in J \/ {b}}
      proof
        let a be object such that
A16:    a in I;
        per cases by A16,XBOOLE_0:def 3;
        suppose
A17:      a in UU;
A18:      J c= J \/ {b} by XBOOLE_1:7;
          ex x being Element of S st a = uparrow x & x in J by A17,A14;
          hence thesis by A18;
        end;
        suppose
A19:      a in {uparrow bb};
A20:      b in {b} by TARSKI:def 1;
A21:      {b} c= J \/ {b} by XBOOLE_1:7;
          a = uparrow bb by A19,TARSKI:def 1;
          hence thesis by A20,A21;
        end;
      end;
      let a be object;
      assume a in {uparrow x where x is Element of S : x in J \/ {b}};
      then consider x being Element of S such that
A22:  a = uparrow x and
A23:  x in J \/ {b};
      per cases by A23,XBOOLE_0:def 3;
      suppose
A24:    x in J;
A25:    UU c= UU \/ {uparrow bb} by XBOOLE_1:7;
        uparrow x in UU by A24,A14;
        hence thesis by A25,A22;
      end;
      suppose
A26:    x in {b};
A27:    {uparrow bb} c= UU \/ {uparrow bb} by XBOOLE_1:7;
A28:    a in {uparrow x} by A22,TARSKI:def 1;
        x = b by A26,TARSKI:def 1;
        hence thesis by A27,A28;
      end;
    end;
A29: (union UU) \/ uparrow bb = union I
    proof
      thus (union UU) \/ uparrow bb c= union I
      proof
        let a be object such that
A30:    a in (union UU) \/ uparrow bb;
        per cases by A30,XBOOLE_0:def 3;
        suppose
A31:      a in union UU;
A32:      UU c= I by XBOOLE_1:7;
          ex A being set st a in A & A in UU by A31,TARSKI:def 4;
          hence thesis by A32,TARSKI:def 4;
        end;
        suppose
A33:      a in uparrow bb;
A34:      uparrow bb in {uparrow bb} by TARSKI:def 1;
          {uparrow bb} c= UU \/ {uparrow bb} by XBOOLE_1:7;
          hence thesis by A34,A33,TARSKI:def 4;
        end;
      end;
      let a be object;
      assume a in union I;
      then consider A being set such that
A35:  a in A and
A36:  A in I by TARSKI:def 4;
      per cases by A36,XBOOLE_0:def 3;
      suppose
        A in UU;
        then A c= union UU by ZFMISC_1:74;
        then
A37:    a in union UU by A35;
        union UU c= (union UU) \/ uparrow bb by XBOOLE_1:7;
        hence thesis by A37;
      end;
      suppose
A38:    A in {uparrow bb};
A39:    uparrow bb c= (union UU) \/ uparrow bb by XBOOLE_1:7;
        A = uparrow bb by A38,TARSKI:def 1;
        hence thesis by A39,A35;
      end;
    end;
    for X being Subset of S st X in UU holds X is upper
    proof
      let X be Subset of S;
      assume X in UU;
      then ex x being Element of S st X = uparrow x & x in J by A14;
      hence thesis;
    end;
    then union UU is upper by WAYBEL_0:28;
    hence thesis by A3,A15,A29,Th28;
  end;
A40: P[{}]
  proof
    deffunc F(Element of S) = (uparrow $1)^0;
    reconsider UU = {} as Subset-Family of S by XBOOLE_1:2;
    take UU;
    reconsider K = union UU as empty Subset of S;
A41: K^0 is empty;
    thus UU = {uparrow x where x is Element of S : x in {}}
    proof
      deffunc F(Element of S) = uparrow $1;
      {F(x) where x is Element of S : x in {}} = {} from EmptySch;
      hence thesis;
    end;
    {F(x) where x is Element of S : x in {}} = {} from EmptySch;
    hence thesis by A41;
  end;
A42: {(uparrow x)^0 where x is Element of S : x in F} = {wayabove x where x
  is Element of S : x in F}
  proof
    thus {(uparrow x)^0 where x is Element of S : x in F} c= {wayabove x where
    x is Element of S : x in F}
    proof
      let a be object;
      assume a in {(uparrow x)^0 where x is Element of S : x in F};
      then consider x being Element of S such that
A43:  a = (uparrow x)^0 and
A44:  x in F;
      (uparrow x)^0 = wayabove x by Th25;
      hence thesis by A43,A44;
    end;
    let a be object;
    assume a in {wayabove x where x is Element of S : x in F};
    then consider x being Element of S such that
A45: a = wayabove x and
A46: x in F;
    (uparrow x)^0 = wayabove x by Th25;
    hence thesis by A45,A46;
  end;
A47: F is finite;
  P[F] from FINSET_1:sch 2(A47,A40,A1);
  then (uparrow F)^0 = union {wayabove x where x is Element of S : x in F} by
A42,YELLOW_9:4;
  hence thesis by Th26;
end;
