reserve A for non degenerated comRing;
reserve R for non degenerated domRing;
reserve n for non empty Ordinal;
reserve o,o1,o2 for object;
reserve X,Y for Subset of Funcs(n,[#]R);
reserve S,T for Subset of Polynom-Ring(n,R);
reserve F,G for FinSequence of the carrier of Polynom-Ring(n,R);
reserve x for Function of n,R;

theorem
    for F be non empty Subset of Ideals(Polynom-Ring(n,R)) holds
    Zero_(union F) = meet{Zero_I where I is Ideal of Polynom-Ring(n,R): I in F}
    proof
      let F be non empty Subset of Ideals(Polynom-Ring(n,R));
set PR = Polynom-Ring(n,R);
set M = {Zero_I where I is Ideal of PR: I in F};
      consider I be object such that
A1:   I in F by XBOOLE_0:def 1;
      I in Ideals(PR) by A1; then
      I in the set of all J where J is Ideal of PR by RING_2:def 3; then
      consider I1 be Ideal of PR such that
A2:   I = I1;
      Zero_I1 in M by A1,A2; then
A3:   M <> {} by XBOOLE_0:def 1;
A4:   for o holds o in Zero_(union F) implies o in meet M
      proof
        let o;
        assume o in Zero_(union F); then
        o in {x where x is Function of n,R : for p be Polynomial of n,R
        st p in union F holds eval(p,x) = 0.R} by Def6; then
        consider x be Function of n,R such that
A6:     x = o &
        for p be Polynomial of n,R st p in union F holds eval(p,x) = 0.R;
        for Z being set holds Z in M implies o in Z
        proof
          let Z be set;
          assume Z in M; then
          consider I be Ideal of PR such that
A8:       Z = Zero_I & I in F;
A9:       I c= union F by A8,ZFMISC_1:74;
          for p1 be Polynomial of n,R st p1 in I holds eval(p1,x) = 0.R
            by A6,A9; then
          x in {x where x is Function of n,R : for p be Polynomial of n,R
          st p in I holds eval(p,x) = 0.R};
          hence thesis by A8, A6,Def6;
        end;
        hence thesis by A3,SETFAM_1:def 1;
      end;
      for o holds o in meet M implies o in Zero_(union F)
      proof
        let o;
        assume
A10:    o in meet M;
        Zero_I1 in M by A1,A2; then
A11:    meet M c= Zero_I1 by SETFAM_1:3;
A12:    meet M c= Funcs(n,[#]R) by A11,XBOOLE_1:1;
        consider x be Function such that
A13:    o = x & dom x = n & rng x c= [#]R by A10,A12,FUNCT_2:def 2;
        reconsider x as Function of n,R by A13,FUNCT_2:2;
        for p be Polynomial of n,R st p in union F holds eval(p,x) = 0.R
        proof
          let p be Polynomial of n,R;
          assume p in union F; then
          consider Ip be set such that
A15:      p in Ip & Ip in F by TARSKI:def 4;
          Ip in Ideals(PR) by A15; then
          Ip in the set of all J where J is Ideal of PR by RING_2:def 3; then
          consider I be Ideal of PR such that
A16:      I = Ip;
          Zero_I in M by A15,A16; then
          o in Zero_I by A3,A10,SETFAM_1:def 1;then
          o in {x where x is Function of n,R : for p be Polynomial of n,R
                st p in I holds eval(p,x) = 0.R} by Def6; then
          consider y be Function of n,R such that
A17:      y = o & for f be Polynomial of n,R st f in I holds eval(f,y)=0.R;
          thus thesis by A17,A15,A16,A13;
        end; then
        x in {x where x is Function of n,R : for p be Polynomial of n,R
              st p in union F holds eval(p,x) = 0.R};
        hence thesis by A13,Def6;
      end;
      hence thesis by TARSKI:2,A4;
    end;
