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 Th13:
    Zero_(0_(n,R)) = Funcs(n,[#]R)
    proof
      o in Funcs(n,[#]R) implies o in Zero_(0_(n,R))
      proof
        assume o in Funcs(n,[#]R); then
        consider x be Function such that
A2:     o = x & dom x = n & rng x c= [#]R by FUNCT_2:def 2;
        reconsider x as Function of n,R by A2,FUNCT_2:2;
        eval(0_(n,R),x) = 0.R by POLYNOM2:20;
        hence thesis by A2;
      end; then
      Funcs(n,[#]R) c= Zero_(0_(n,R));
      hence thesis;
    end;
