 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 Th9:
  for f,J st not f in sqrt J holds
    ex m be prime Ideal of A st not f in m & J c= m
    proof
      let f,J;
      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;
      P partially_orders S by WELLORD2:19,21,20; then
      consider I being set such that
A3:   I is_maximal_in P by A1,A2,Th8,ORDERS_1:63;
      I in S by WELLORD2:def 1, A3; then
      consider p be Subset of A such that
A4:   p = I and
A6:   p is proper Ideal of A and
A7:   J c= p and
A8:   p /\ multClSet(J,f) = {};
      f|^1 = f by BINOM:8; then
      f in multClSet(J,f); then
A11:  not f in p by A8,XBOOLE_0:def 4;
      p is quasi-prime Ideal of A
      proof
        for x,y be Element of A st not x in p & not y in p holds not x*y in p
        proof
          let x,y be Element of A;
          assume that
A12:      not x in p and
A13:      not y in p;
          assume
A14:      x*y in p;
A15:      p c< p + {x}-Ideal
          proof
            {x}-Ideal c= p + {x}-Ideal by A6,IDEAL_1:74;
            hence thesis by A6,A12,IDEAL_1:66,73;
          end;
A17:      p c< p + {y}-Ideal
          proof
            {y}-Ideal c= p + {y}-Ideal by A6,IDEAL_1:74;
            hence thesis by A6,A13,IDEAL_1:66,73;
          end;
          set J2 = p + {x}-Ideal;
A19:      not (p + {x}-Ideal in Ideals(A,J,f) )
          proof
            assume
A20:        p + {x}-Ideal in Ideals(A,J,f); then
A21:        J2 in field P by WELLORD2:def 1;
            J2 <> I & [I,J2] in P by A2,A3,A4,A15,A20,WELLORD2:def 1;
            hence contradiction by A3,A21;
          end;
A22:      not (p + {y}-Ideal in Ideals(A,J,f) )
          proof
            assume
A23:        p + {y}-Ideal in Ideals(A,J,f);
            set J2 = p + {y}-Ideal;
A24:        J2 in field P by WELLORD2:def 1,A23;
            J2 <> I & [I,J2] in P by A2,A3,A4,A17,A23,WELLORD2:def 1;
            hence contradiction by A3,A24;
          end;
          reconsider q = p + {x}-Ideal as Subset of A;
A26:      p c= p + {x}-Ideal by A6,IDEAL_1:73;
A27:      not(q is proper Ideal of A) or not (J c= q) or
          not (q /\ multClSet(J,f) = {}) by A19;
A28:      ex n be Nat st f|^n in p + {x}-Ideal
          proof
            per cases by A7,A26,A27;
              suppose
A29:            not(p + {x}-Ideal is proper Ideal of A);
                reconsider q1 = p + {x}-Ideal as Ideal of A by A6;
A30:            q1 = the carrier of A by A29, SUBSET_1:def 6;
                reconsider n = 1 as Nat;
                f|^n in q by A30;
                hence thesis;
              end;
              suppose not (q /\ multClSet(J,f) = {}); then
                consider h be object such that
A32:            h in q /\ multClSet(J,f) by XBOOLE_0:def 1;
                h in q & h in multClSet(J,f) by A32,XBOOLE_0:def 4;
                then consider n1 be Nat such that
A33:            h = f|^n1;
                f|^n1 in q by A33,A32,XBOOLE_0:def 4;
                hence thesis;
              end;
          end;
          reconsider q = p + {y}-Ideal as Subset of A;
          p c= p + {y}-Ideal by A6,IDEAL_1:73; then
A36:      J c= q by A7;
A37:      ex m be Nat st f|^m in p + {y}-Ideal
          proof
            per cases by A22,A36;
              suppose
A38:            not(p + {y}-Ideal is proper Ideal of A);
                reconsider q1 = p + {y}-Ideal as Ideal of A by A6;
A39:            q1 = the carrier of A by A38,SUBSET_1:def 6;
                reconsider m = 1 as Nat;
                f|^m in q by A39;
                hence thesis;
              end;
              suppose q /\ multClSet(J,f) <> {}; then
                consider h be object such that
A41:            h in q /\ multClSet(J,f) by XBOOLE_0:def 1;
                h in q & h in multClSet(J,f) by A41,XBOOLE_0:def 4;
                then
                consider n1 be Nat such that
A42:            h = f|^n1;
                f|^n1 in q by A41,XBOOLE_0:def 4,A42;
                hence thesis;
              end;
          end;
          reconsider p as Ideal of A by A6;
          consider n be Nat such that
A43:      f|^n in p + {x}-Ideal by A28;
          consider m be Nat such that
A44:      f|^m in p + {y}-Ideal by A37;
          f|^n in {a + b where a,b is Element of A :
          a in p & b in {x}-Ideal} by A43,IDEAL_1:def 19; then
          consider p1,x1 be Element of A such that
A45:      f|^n = p1+x1 and
A46:      p1 in p and
A47:      x1 in {x}-Ideal;
          f|^m in {a + b where a,b is Element of A :
          a in p & b in {y}-Ideal} by A44,IDEAL_1:def 19; then
          consider p2,y2 be Element of A such that
A48:      f|^m = p2+y2 and
A49:      p2 in p and
A50:      y2 in {y}-Ideal;
          x1 in the set of all x*a where a is Element of A by A47,IDEAL_1:64;
          then
          consider a be Element of A such that
A51:      x1 = x*a;
          y2 in the set of all y*a where a is Element of A by A50,IDEAL_1:64;
          then
          consider b be Element of A such that
A52:      y2 = y*b;
A53:      (p1 + x1) * p2 + (p1 * y2) in p
          proof
            reconsider a = p1 + x1 as Element of A;
A54:        a * p2 in p by A49,IDEAL_1:def 2;
            p1 * y2 in p by A46,IDEAL_1:def 3;
            hence thesis by A54,IDEAL_1:def 1;
          end;
A56a:     x1 * y2 in {x*y}-Ideal
          proof
A57:        x1 * y2 = a*(x*(y*b)) by GROUP_1:def 3,A52,A51
              .= ((x*y)*b)*a by GROUP_1:def 3
              .= (x*y)*(b*a) by GROUP_1:def 3;
            (x*y)*(b*a) in the set of all (x*y)*r where r is Element of A;
            hence thesis by A57,IDEAL_1:64;
          end;
A59:      f|^(n+m) = (f|^n) * (f|^m) by BINOM:10
          .= (p1 + x1) * p2 + ((p1 + x1) * y2) by VECTSP_1:def 7,A45,A48
          .= (p1 + x1) * p2 + (p1 * y2 + x1 * y2) by VECTSP_1:def 7
          .= ((p1 + x1) * p2 + (p1 * y2)) + (x1 * y2) by RLVECT_1:def 3;
        reconsider s = (p1 + x1) * p2 + (p1 * y2), t = x1 * y2 as Element of A;
          s+t in {u+v where u,v is Element of A:u in p & v in {x*y}-Ideal}
          by A53,A56a; then
A61:      f|^(n+m) in p + {x*y}-Ideal by A59,IDEAL_1:def 19;
          reconsider n1 = n + m as Nat;
A63:      f|^(n+m) in multClSet(J,f);
A64:      not p + {x*y}-Ideal in Ideals(A,J,f)
          proof
            assume p + {x*y}-Ideal in Ideals(A,J,f); then
            consider q be Subset of A such that
A66:        q = p + {x*y}-Ideal and
            q is proper Ideal of A & J c= q and
A67:        q /\ multClSet(J,f) = {};
            thus contradiction by A61,A63,XBOOLE_0:def 4,A67,A66;
          end;
          p-Ideal = p by IDEAL_1:44; then
          {x*y}-Ideal c= p by IDEAL_1:67, A14; then
          p = p + {x*y}-Ideal by IDEAL_1:74,75;
          hence contradiction by A3,A4,A64,WELLORD2:def 1;
        end; then
        for a, b being Element of A st a*b in p holds a in p or b in p;
        hence thesis by A6,RING_1:def 1;
      end;
      hence thesis by A6,A7,A11;
    end;
