 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;

theorem Th8:
  for A, J, f st not f in sqrt J holds
    Ideals(A,J,f) has_upper_Zorn_property_wrt RelIncl(Ideals(A,J,f) )
    proof
      let A,J,f;
      assume
A1:   not f in sqrt J;
      set S = Ideals(A,J,f);
      set P = RelIncl(S);
A2:   field P = S by WELLORD2:def 1;
      for Y be set st Y c= S & P|_2 Y is being_linear-order
      ex x be set st x in S & for y be set st y in Y holds [y,x] in P
      proof
        let Y be set such that
A3:     Y c= S and
A4:     P |_2 Y is being_linear-order;
        per cases;
          suppose
A5:         Y is empty;
            take x = J;
            thus thesis by A5,A1,Th7;
          end;
          suppose
            Y is non empty; then
            consider e being object such that
A6:         e in Y;
            take x = union Y;
            for a be object st a in x holds a in the carrier of A
            proof
              let a be object;
              assume a in x; then
              consider Z being set such that
A7:           a in Z and
A8:           Z in Y by TARSKI:def 4;
              Z in S by A3,A8; then
              ex A1 be Subset of A st Z = A1 &
              A1 is proper Ideal of A & J c= A1 & A1 /\ multClSet(J,f) = {};
              hence thesis by A7;
            end; then
            x c= the carrier of A; then
            reconsider B = x as Subset of A;
A9:         B is left-ideal
            proof
              let p, a be Element of A;
              assume a in B; then
              consider Aa being set such that
A10:          a in Aa and
A11:          Aa in Y by TARSKI:def 4;
              Aa in S by A3,A11; then
              consider Ia be Subset of A such that
A12:          Aa = Ia and
A13:          Ia is proper Ideal of A and
              J c= Ia and
              Ia /\ multClSet(J,f) = {};
              p*a in Ia & Ia c= B by A10,A11,A12,A13,IDEAL_1:def 2,ZFMISC_1:74;
              hence thesis;
            end;
A14:        B is proper
            proof
              assume B is non proper; then
              consider Aa being set such that
A16:          1.A in Aa and
A17:          Aa in Y by TARSKI:def 4;
              Aa in S by A3,A17; then
              ex Ia be Subset of A st Aa = Ia &
              Ia is proper Ideal of A & J c= Ia & Ia /\ multClSet(J,f) = {};
              hence contradiction by A16,IDEAL_1:19;
            end;
A18:        B is add-closed
            proof
A19:          field (P |_2 Y) = Y by A2,A3,ORDERS_1:71;
              let a, b be Element of A;
A20:          now
              let A1 be Ideal of A;
              assume a in A1 & b in A1;
              then
A21:          a+b in A1 by IDEAL_1:def 1;
              assume A1 in Y;
              hence a+b in B by A21,TARSKI:def 4;
            end;
            assume a in B; then
            consider Aa being set such that
A22:        a in Aa and
A23:        Aa in Y by TARSKI:def 4;
            Aa in S by A3,A23; then
A24:        ex Ia be Subset of A st Aa = Ia &
            Ia is proper Ideal of A & J c= Ia & Ia /\ multClSet(J,f) = {};
            assume b in B; then
            consider Ab being set such that
A25:        b in Ab and
A26:        Ab in Y by TARSKI:def 4;
            [Aa,Ab] in P |_2 Y or [Ab,Aa] in P |_2 Y or Aa = Ab
            by A4, A23, A26, A19, RELAT_2:def 6,def 14; then
            [Aa,Ab] in P or [Ab,Aa] in P or Aa = Ab by XBOOLE_0:def 4; then
A27:        Aa c= Ab or Ab c= Aa by A3,A23,A26,WELLORD2:def 1;
            Ab in S by A3,A26; then
            ex Ib be Subset of A st Ab = Ib &
            Ib is proper Ideal of A & J c= Ib & Ib /\ multClSet(J,f) = {};
            hence a+b in B by A22,A23,A25,A26,A20,A24,A27;
          end;
          e in S by A3,A6; then
          consider A1 be Subset of A such that
A28:      A1 = e and
A29:      A1 is proper Ideal of A and
A30:      J c= A1 and
          A1 /\ multClSet(J,f) = {};
          ex q being object st q in A1 by XBOOLE_0:def 1,A29; then
A32:      B is non empty & J c= B by A6,A28,A30,TARSKI:def 4;
A33:      B /\ multClSet(J,f) = {}
          proof
            assume B /\ multClSet(J,f) <> {}; then
            consider y be object such that
A35:        y in B /\ multClSet(J,f) by XBOOLE_0:def 1;
            y in B & y in multClSet(J,f) by A35,XBOOLE_0:def 4; then
            consider Aa being set such that
A36:        y in Aa and
A37:        Aa in Y by TARSKI:def 4;
            Aa in S by A3,A37; then
            consider Ia be Subset of A such that
A38:        Aa = Ia and
            Ia is proper Ideal of A & J c= Ia and
A39:        Ia /\ multClSet(J,f) = {};
            y in multClSet(J,f) by A35, XBOOLE_0:def 4;
            hence contradiction by A39, A38, A36, XBOOLE_0:def 4;
          end;
          thus
A41:      x in S by A9,A14,A18,A32,A33;
          let y be set;
          assume
A42:      y in Y; then
          y c= x by ZFMISC_1:74;
          hence thesis by A3,A41,A42,WELLORD2:def 1;
        end;
      end;
      hence thesis;
    end;
