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 Th19:
    for S,T be Algebraic_Set of n,R holds S /\ T is Algebraic_Set of n,R
    proof
      let S,T be Algebraic_Set of n,R;
      consider I be Ideal of Polynom-Ring(n,R) such that
A1:   S = Zero_(I) by Def7;
      consider J be Ideal of Polynom-Ring(n,R) such that
A2:   T = Zero_(J) by Def7;
      reconsider Z = I \/ J as non empty Subset of Polynom-Ring(n,R);
      reconsider M = Z-Ideal as Ideal of Polynom-Ring(n,R);
      S /\ T = Zero_(I \/ J) by A1,A2,Th18
      .= Zero_(Z-Ideal) by Th17; then
      S /\ T is algebraic_set_from_ideal;
      hence thesis;
    end;
