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 m be non zero Nat, P be Subset of singletons(Funcs(n,[#]R)) holds
    card P = m implies union P is Algebraic_Set of n,R
    proof
      let m be non zero Nat, P be Subset of singletons(Funcs(n,[#]R));
      assume
A1:   card P = m;
A2:   singletons(Funcs(n,[#]R)) c= Alg_Sets(n,R)
      proof
        for o holds o in singletons(Funcs(n,[#]R)) implies o in Alg_Sets(n,R)
        proof
          let o;
          assume o in singletons(Funcs(n,[#]R)); then
          o in { S where S is Subset of Funcs(n,[#]R) : S is 1-element }
            by BSPACE:def 8; then
          consider S be Subset of Funcs(n,[#]R) such that
A4:       o = S & S is 1-element;
          consider p be Element of Funcs(n,[#]R) such that
A5:       S = {p} by A4,CARD_1:65;
          S is Algebraic_Set of n,R by A5,Th26;
          hence thesis by A4;
        end;
        hence thesis;
      end;
      P c= Alg_Sets(n,R) by A2;
      hence thesis by A1,Th24;
    end;
