reserve A for RelStr;
reserve X for non empty set;
reserve PX,PY,PZ,Y,a,b,c,x,y for set;
reserve S1,S2 for Subset of Y;

theorem Th16:
  for H being covering T_3 Hierarchy of Y st H is lower-bounded &
not {} in H for A being Subset of Y st A in H ex P being a_partition of Y st A
  in P & P c= H
proof
  let H be covering T_3 Hierarchy of Y such that
A1: H is lower-bounded and
A2: not {} in H;
  let A be Subset of Y such that
A3: A in H;
  set k1 = {A};
A4: k1 c= H by A3,ZFMISC_1:31;
A5: A in k1 by TARSKI:def 1;
A6: k1 c= bool Y by ZFMISC_1:31;
  defpred X[set] means A in $1 & $1 is mutually-disjoint & $1 c= H;
  consider K be set such that
A7: for S be set holds S in K iff S in bool bool Y & X[S] from XFAMILY:
  sch 1;
  k1 is mutually-disjoint by Th9;
  then
A8: k1 in K by A7,A5,A4,A6;
  for Z be set st Z c= K & Z is c=-linear ex X3 be set st X3 in K & for
  X1 be set st X1 in Z holds X1 c= X3
  proof
    let Z be set such that
A9: Z c= K and
A10: Z is c=-linear;
A11: for X1,X2 being set st X1 in Z & X2 in Z holds X1 c= X2 or X2 c= X1
        by XBOOLE_0:def 9,A10,ORDINAL1:def 8;
    per cases;
    suppose
A12:  Z <> {};
      set X3 = union Z;
      take X3;
      now
        consider z be object such that
A13:    z in Z by A12,XBOOLE_0:def 1;
        reconsider z as set by TARSKI:1;
        A in z by A7,A9,A13;
        hence A in X3 by A13,TARSKI:def 4;
A14:    for a st a in Z holds a c= H by A7,A9;
        then X3 c= H by ZFMISC_1:76;
        then X3 c= bool Y by XBOOLE_1:1;
        hence X3 in bool bool Y;
        thus X3 is mutually-disjoint
        proof
          let t1,t2 be set such that
A15:      t1 in X3 and
A16:      t2 in X3 and
A17:      t1 <> t2;
          consider v1 be set such that
A18:      t1 in v1 and
A19:      v1 in Z by A15,TARSKI:def 4;
A20:      v1 is mutually-disjoint by A7,A9,A19;
          consider v2 be set such that
A21:      t2 in v2 and
A22:      v2 in Z by A16,TARSKI:def 4;
A23:      v2 is mutually-disjoint by A7,A9,A22;
          per cases by A11,A19,A22;
          suppose
            v1 c= v2;
            hence thesis by A17,A18,A21,A23;
          end;
          suppose
            v2 c= v1;
            hence thesis by A17,A18,A21,A20;
          end;
        end;
        thus X3 c= H by A14,ZFMISC_1:76;
      end;
      hence X3 in K by A7;
      thus thesis by ZFMISC_1:74;
    end;
    suppose
A24:  Z = {};
      consider a be set such that
A25:  a in K by A8;
      take a;
      thus a in K by A25;
      thus thesis by A24;
    end;
  end;
  then consider M be set such that
A26: M in K and
A27: for Z be set st Z in K & Z <> M holds not M c= Z by A8,ORDERS_1:65;
A28: M is mutually-disjoint Subset-Family of Y by A7,A26;
A29: for C being mutually-disjoint Subset-Family of Y
      st A in C & C c= H & M c= C
    holds M = C by A27,A7;
A30: A in M by A7,A26;
  take M;
  M c= H by A7,A26;
  hence thesis by A1,A2,A28,A30,A29,Th15;
end;
