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 Th17:
    for S be non empty Subset of Polynom-Ring(n,R) holds
    Zero_(S) = Zero_(S-Ideal)
    proof
      let S be non empty Subset of Polynom-Ring(n,R);
A1:   S c= S-Ideal by IDEAL_1:def 14;
      Zero_(S) c= Zero_(S-Ideal)
      proof
set MS = {x where x is Function of n,R : for f be Polynomial of n,R
          st f in S holds eval(f,x) = 0.R};
set MI = {x where x is Function of n,R : for f be Polynomial of n,R
          st f in S-Ideal holds eval(f,x) = 0.R};
        for o holds o in Zero_(S) implies o in Zero_(S-Ideal)
        proof
          let o;
          assume o in Zero_(S); then
          o in MS by Def6; then
          consider x be Function of n,R such that
A3:       o = x & for f be Polynomial of n,R
          st f in S holds eval(f,x) = 0.R;
          for p be Polynomial of n,R st p in S-Ideal holds eval(p,x) = 0.R
          proof
            let p be Polynomial of n,R;
            assume p in S-Ideal; then
            consider F be LinearCombination of S such that
A5:         p = Sum F by IDEAL_1:60;
A6:         for i being Nat st i in dom E_eval(F,x) holds E_eval(F,x).i = 0.R
            proof
              let i be Nat;
              assume i in dom E_eval(F,x); then
A8:           i in dom F by Def2;
              consider u,v being Element of Polynom-Ring(n,R),
              a being Element of S such that
A9:          F/.i = u*a*v by A8,IDEAL_1:def 8;
A10:          a in S by SUBSET_1:def 1;
              a in Polynom-Ring(n,R) by SUBSET_1:def 1; then
            reconsider a1 = a as Polynomial of n,R by POLYNOM1:def 11;
A11:          E_eval(a,x) = eval(a1,x) by Def1 .= 0.R by A10,A3;
              F/.i = (u*v)*a by A9,GROUP_1:def 3; then
              E_eval(F,x).i = E_eval((u*v)*a,x) by A8,Def2
              .= E_eval(u*v,x) * 0.R by A11,Th3 .= 0.R;
              hence thesis;
            end;
A13:        Sum E_eval(F,x) = Sum (len(E_eval(F,x))|-> 0.R) by A6,Lm1
            .= 0.R by Lm2;
            eval(p,x) = E_eval(Sum F,x) by A5,Def1
            .= 0.R by A13,Th5;
            hence thesis;
          end; then
          x in MI;
          hence thesis by A3,Def6;
        end;
        hence Zero_(S) c= Zero_(S-Ideal);
      end;
      hence Zero_(S) = Zero_(S-Ideal) by A1,Th16;
    end;
