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 Hierarchy of Y st H is with_max's ex P being
  a_partition of Y st P c= H
proof
  let H be covering Hierarchy of Y such that
A1: H is with_max's;
  per cases;
  suppose
A2: Y = {};
    set P9 = {};
    reconsider P = P9 as Subset-Family of Y by XBOOLE_1:2;
    take P;
    thus thesis by A2,EQREL_1:45;
  end;
  suppose
A3: Y <> {};
    now
      per cases;
      suppose
A4:     Y in H;
        set P = {Y};
A5:     P c= H
        by A4,TARSKI:def 1;
        P is a_partition of Y by A3,EQREL_1:39;
        hence thesis by A5;
      end;
      suppose
A6:     not Y in H;
        defpred X[set] means ex S be Subset of Y st (S in H & S c= $1 & for V
        being Subset of Y st $1 c= V & V in H holds V = Y);
        consider P9 be set such that
A7:     for T be set holds T in P9 iff T in H & X[T] from XFAMILY:
        sch 1;
A8:     P9 c= H
        by A7;
        then reconsider P = P9 as Subset-Family of Y by XBOOLE_1:1;
A9:     now
          let S1 such that
A10:      S1 in P;
          thus S1 <> {}
          proof
            consider S be Subset of Y such that
A11:        S in H and
A12:        S c= S1 and
A13:        for V being Subset of Y st S1 c= V & V in H holds V = Y by A7,A10;
            assume
A14:        S1 = {};
            then S1 = S by A12
              .= Y by A14,A11,A13,XBOOLE_1:2;
            hence contradiction by A3,A14;
          end;
          thus for S2 st S2 in P holds S1 = S2 or S1 misses S2
          proof
            let S2 such that
A15:        S2 in P;
A16:        ex T be Subset of Y st T in H & T c= S2 & for V being
            Subset of Y st S2 c= V & V in H holds V = Y by A7,A15;
            S2 in H by A7,A15;
            hence thesis by A6,A16;
          end;
        end;
A17:    union H = Y by ABIAN:4;
        Y c= union P
        proof
          let p be object;
          assume p in Y;
          then consider h9 be set such that
A18:      p in h9 and
A19:      h9 in H by A17,TARSKI:def 4;
          reconsider h = h9 as Subset of Y by A19;
          consider T be Subset of Y such that
A20:      h c= T and
A21:      T in H and
A22:      for V being Subset of Y st T c= V & V in H holds V = Y by A1,A19;
          T in P by A7,A21,A22;
          hence thesis by A18,A20,TARSKI:def 4;
        end;
        then union P = Y by XBOOLE_0:def 10;
        then P is a_partition of Y by A9,EQREL_1:def 4;
        hence thesis by A8;
      end;
    end;
    hence thesis;
  end;
end;
