reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem Th11:
  for y being Element of Y
  ex X being Subset of Y st y in X & X is_min_depend PA,PB
proof
  let y be Element of Y;
A1: union PA = Y by EQREL_1:def 4;
A2: for A being set st A in PA holds A<>{} & for B being set st B in PA
  holds A=B or A misses B by EQREL_1:def 4;
A3: Y is_a_dependent_set_of PA & Y is_a_dependent_set_of PB by Th7;
  defpred P[set] means
  y in $1 & $1 is_a_dependent_set_of PA & $1 is_a_dependent_set_of PB;
  reconsider XX={X where X is Subset of Y:P[X]}
  as Subset-Family of Y from DOMAIN_1:sch 7;
  reconsider XX as Subset-Family of Y;
 Y c= Y;
then A4: Y in XX by A3;
 for X1 be set st X1 in XX holds y in X1
  proof
    let X1 be set;
    assume X1 in XX;
then  ex X be Subset of Y st X=X1 & y in X & X
    is_a_dependent_set_of PA & X is_a_dependent_set_of PB;
    hence thesis;
  end;
then A5: y in meet XX by A4,SETFAM_1:def 1;
then A6: Intersect(XX)<>{} by SETFAM_1:def 9;
  take Intersect(XX);
 for X1 be set st X1 in XX holds X1 is_a_dependent_set_of PA
  proof
    let X1 be set;
    assume X1 in XX;
then  ex X be Subset of Y st X=X1 & y in X & X is_a_dependent_set_of PA &
    X is_a_dependent_set_of PB;
    hence thesis;
  end;
then A7: Intersect(XX) is_a_dependent_set_of PA by A6,Th8;
 for X1 be set st X1 in XX holds X1 is_a_dependent_set_of PB
  proof
    let X1 be set;
    assume X1 in XX;
then  ex X be Subset of Y st X=X1 & y in X & X is_a_dependent_set_of PA &
    X is_a_dependent_set_of PB;
    hence thesis;
  end;
then A8: Intersect(XX) is_a_dependent_set_of PB by A6,Th8;
 for d being set st d c= Intersect(XX) & d is_a_dependent_set_of PA &
  d is_a_dependent_set_of PB holds d=Intersect(XX)
  proof
    let d be set;
    assume that
A9: d c= Intersect(XX) and
A10: d is_a_dependent_set_of PA and
A11: d is_a_dependent_set_of PB;
    consider Ad being set such that
A12: Ad c= PA and
A13: Ad<>{} and
A14: d = union Ad by A10;
A15: d c= Y by A1,A12,A14,ZFMISC_1:77;
    per cases;
    suppose
  y in d;
then A16:  d in XX by A10,A11,A15;
  Intersect(XX) c= d
      proof
        let X1 be object;
        assume X1 in Intersect(XX);
then     X1 in meet XX by A4,SETFAM_1:def 9;
        hence thesis by A16,SETFAM_1:def 1;
      end;
      hence thesis by A9,XBOOLE_0:def 10;
    end;
    suppose
A17:  not y in d;
      reconsider d as Subset of Y by A1,A12,A14,ZFMISC_1:77;
  d` = Y \ d by SUBSET_1:def 4;
then A18:  y in d` by A17,XBOOLE_0:def 5;
  d misses d` by SUBSET_1:24;
then A19:  d /\ d` = {} by XBOOLE_0:def 7;
  d <> Y by A17;
      then   d
` is_a_dependent_set_of PA & d` is_a_dependent_set_of PB by A10,A11,Th10;
then A20:  d` in XX by A18;
  Intersect(XX) c= d`
      proof
        let X1 be object;
        assume X1 in Intersect(XX);
then     X1 in meet XX by A4,SETFAM_1:def 9;
        hence thesis by A20,SETFAM_1:def 1;
      end;
then A21:  d /\ Intersect(XX) = {} by A19,XBOOLE_1:3,26;
  d /\ d c= d /\ Intersect(XX) by A9,XBOOLE_1:26;
then A22:  union Ad = {} by A14,A21;
      consider ad being object such that
A23:  ad in Ad by A13,XBOOLE_0:def 1;
A24:  ad<>{} by A2,A12,A23;
      reconsider ad as set by TARSKI:1;
  ad c= {} by A22,A23,ZFMISC_1:74;
      hence thesis by A24;
    end;
  end;
  hence thesis by A4,A5,A7,A8,SETFAM_1:def 9;
end;
