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
  for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {}
  in H for A being Subset of Y for B being mutually-disjoint Subset-Family of Y
st A in B & B c= H & for C being mutually-disjoint Subset-Family of Y st A in C
& C c= H & union B c= union C holds union B = union C holds B is a_partition of
  Y
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;
  let B be mutually-disjoint Subset-Family of Y such that
A3: A in B and
A4: B c= H and
A5: for C being mutually-disjoint Subset-Family of Y st A in C & C c= H
  & union B c= union C holds union B = union C;
A6: union H = Y by ABIAN:4;
A7: H is hierarchic by Def4;
  per cases;
  suppose
A8: Y <> {};
    Y c= union B
    proof
      assume not Y c= union B;
      then consider x be object such that
A9:   x in Y and
A10:  not x in union B;
      consider xx be set such that
A11:  x in xx and
A12:  xx in H by A6,A9,TARSKI:def 4;
      defpred X[set] means x in $1;
      consider D be set such that
A13:  for h be set holds h in D iff h in H & X[h] from XFAMILY:sch 1;
A14:  D c= H
      by A13;
      now
        let h1,h2 be set such that
A15:    h1 in D and
A16:    h2 in D;
        now
A17:      x in h2 by A13,A16;
          assume
A18:      h1 misses h2;
          x in h1 by A13,A15;
          hence contradiction by A18,A17,XBOOLE_0:3;
        end;
        then h1 c= h2 or h2 c= h1 by A7,A14,A15,A16;
        hence h1,h2 are_c=-comparable by XBOOLE_0:def 9;
      end;
      then
A19:  D is c=-linear by ORDINAL1:def 8;
      xx in D by A13,A11,A12;
      then consider min1 be set such that
A20:  min1 in H and
A21:  min1 c= meet D by A1,A14,A19;
      reconsider min9 = min1 as Subset of Y by A20;
      set C = B \/ {min9};
A22:  B c= C by XBOOLE_1:7;
      now
        reconsider x9 = x as Element of Y by A9;
        let b1 be set such that
A23:    b1 in B;
        reconsider b19 = b1 as Subset of Y by A23;
A24:    not x9 in b19 by A10,A23,TARSKI:def 4;
A25:    not b1 c= min9
        proof
          reconsider b19 = b1 as Subset of Y by A23;
          consider k be Subset of Y such that
A26:      x9 in k and
A27:      k in H and
A28:      k misses b19 by A4,A23,A24,Def6;
          k in D by A13,A26,A27;
          then meet D c= k by SETFAM_1:3;
          then
A29:      min9 c= k by A21;
          b1 <> {} by A2,A4,A23;
          then
A30:      ex y be object st y in b19 by XBOOLE_0:def 1;
          assume b1 c= min9;
          then b19 c= k by A29;
          hence contradiction by A28,A30,XBOOLE_0:3;
        end;
        not min9 c= b1
        proof
          consider k be Subset of Y such that
A31:      x9 in k and
A32:      k in H and
A33:      k misses b19 by A4,A23,A24,Def6;
          k in D by A13,A31,A32;
          then meet D c= k by SETFAM_1:3;
          then
A34:      min9 c= k by A21;
          assume
A35:      min9 c= b1;
          ex y be object st y in min9 by A2,A20,XBOOLE_0:def 1;
          hence contradiction by A35,A33,A34,XBOOLE_0:3;
        end;
        hence b1 misses min9 by A4,A7,A20,A23,A25;
      end;
      then
A36:  for b be set st b in B holds min9 misses b & Y <> {} by A8;
      then
A37:  C is mutually-disjoint Subset-Family of Y by Th11;
      {min9} c= H
      by A20,TARSKI:def 1;
      then
A38:  C c= H by A4,XBOOLE_1:8;
      union C <> union B by A2,A20,A36,Th11;
      hence contradiction by A3,A5,A37,A38,A22,ZFMISC_1:77;
    end;
    then
A39: union B = Y by XBOOLE_0:def 10;
    for A be Subset of Y st A in B holds (A <> {} & for E be Subset of Y
    st E in B holds A = E or A misses E) by A2,A4,Def5;
    hence thesis by A39,EQREL_1:def 4;
  end;
  suppose
A40: Y = {};
    now
      per cases by A40,ZFMISC_1:1,33;
      suppose
        H = {};
        hence B = {} by A4;
      end;
      suppose
A41:    H = {{}};
        B <> {{}}
        proof
          assume B = {{}};
          then {} in B by TARSKI:def 1;
          hence contradiction by A2,A4;
        end;
        hence B = {} by A4,A41,ZFMISC_1:33;
      end;
    end;
    hence thesis by A40,EQREL_1:45;
  end;
end;
