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 Th25:
    for a be Function of n,R holds Zero_(polyset(a)) = {a}
    proof
      let a be Function of n,R;
      thus o in Zero_(polyset(a)) implies o in {a}
      proof
        assume
A2:     o in Zero_(polyset(a)); then
        consider x be Function such that
A4:     o = x & dom x = n & rng x c= [#]R by FUNCT_2:def 2;
        assume
A3:     not o in {a};
        reconsider x as Function of n,R by A4,FUNCT_2:2;
A5:     dom a = n by FUNCT_2:def 1 .= dom x by FUNCT_2:def 1;
        x <> a by A4,A3,TARSKI:def 1; then
        consider i be object such that
A6:     i in dom a & x.i <> a.i by A5;
        reconsider i0 = i as Element of n by A6;
        reconsider f0 = deg1Poly(a,i0) as Polynomial of n,R;
A7:     f0 in polyset(a);
A8:     eval(f0,x) = eval(1_1(i0,R),x) - eval((a.i0)|(n,R),x) by POLYNOM2:24
        .= x.i0 - eval((a.i0)|(n,R),x) by HILB10_3:1
        .= x.i0 - (a.i0) by POLYNOM7:25;
        not x in Zero_(polyset(a))
        proof
          assume x in Zero_(polyset(a)); then
          x in {x where x is Function of n,R : for p be Polynomial of n,R
               st p in polyset(a) holds eval(p,x) = 0.R} by Def6; then
          consider x0 be Function of n,R such that
A10:      x0 = x & for f be Polynomial of n,R
          st f in polyset(a) holds eval(f,x0) = 0.R;
          thus contradiction by RLVECT_1:21,A6,A8,A7,A10;
        end;
        hence contradiction by A2,A4;
      end;
      thus o in {a} implies o in Zero_(polyset(a))
      proof
        assume o in {a}; then
A12:    o = a by TARSKI:def 1;
        for f be Polynomial of n,R st f in polyset(a) holds  eval(f,a) = 0.R
        proof
          let f be Polynomial of n,R;
          assume f in polyset(a); then
          consider f0 be Polynomial of n,R such that
A14:      f0 = f & ex i be Element of n st f0 = deg1Poly(a,i);
          consider i0 be Element of n such that
A15:      f0 = deg1Poly(a,i0) by A14;
          eval(f0,a)
          = eval(1_1(i0,R),a) - eval((a.i0)|(n,R),a) by A15,POLYNOM2:24
          .= a.i0 - eval((a.i0)|(n,R),a) by HILB10_3:1
          .= a.i0 - a.i0 by POLYNOM7:25
          .= 0.R by RLVECT_1:15;
          hence thesis by A14;
        end; then
        a in {x where x is Function of n,R :
        for f be Polynomial of n,R st f in polyset(a) holds eval(f,x)=0.R};
        hence thesis by A12, Def6;
      end;
    end;
