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 Th24:
    for m be non zero Nat, F be Subset of Alg_Sets(n,R) holds
    card F = m implies union F is Algebraic_Set of n,R
    proof
      let m be non zero Nat, F be Subset of Alg_Sets(n,R);
      defpred P[Nat] means
      for G be Subset of Alg_Sets(n,R) holds
      card G = $1 implies union G is Algebraic_Set of n,R;
A1:   for m be non zero Nat st P[m] holds P[(m+1)]
      proof
        let m be non zero Nat;
        assume
A2:     P[m];
        for G be Subset of Alg_Sets(n,R) holds
        card G = m+1 implies union G is Algebraic_Set of n,R
        proof
          let G be Subset of Alg_Sets(n,R);
          assume
A3:       card G = m+1; then
          G <> {}; then
          consider o such that
A4:       o in G by XBOOLE_0:def 1;
          o in {S where S is Subset of [#]Funcs(n,[#]R):
          S is Algebraic_Set of n,R } by A4; then
          consider S be Subset of [#]Funcs(n,[#]R) such that
A5:       o = S & S is Algebraic_Set of n,R;
A6:       G is finite by A3;
  reconsider G1 = G \ {S} as finite Subset of Alg_Sets(n,R) by A6;
A7:       not S in G1 by ZFMISC_1:56;
A8:       m+1 = card (G1 \/ {S}) by A3,A4,A5,ZFMISC_1:31,FIELD_5:1
          .= card G1 + 1 by A7,CARD_2:41;
          union G1 is Algebraic_Set of n,R by A2,A8; then
          consider I be Ideal of Polynom-Ring(n,R) such that
A9:       union G1 = Zero_(I) by Def7;
          consider J be Ideal of Polynom-Ring(n,R) such that
A10:      S = Zero_(J) by A5,Def7;
A11:      Zero_(I /\ J) = union G1 \/ S by A9,A10,Th22
          .= union G1 \/ union {S} by ZFMISC_1:25
          .= union (G1 \/ {S}) by ZFMISC_1:78;
          consider I1 be Ideal of Polynom-Ring(n,R) such that
A12:      I1 = I /\ J;
          Zero_I1 = union G by A11,A4,A5,ZFMISC_1:31,FIELD_5:1,A12;
          hence thesis by Def7;
        end;
        hence thesis;
      end;
A14:  P[1]
      proof
        for G be Subset of Alg_Sets(n,R) holds
        card G = 1 implies union G is Algebraic_Set of n,R
        proof
          let G be Subset of Alg_Sets(n,R);
          assume card G = 1; then
          consider o being object such that
A16:      G = {o} by CARD_2:42;
          o in G by A16,TARSKI:def 1; then
          o in {S where S is Subset of Funcs(n,[#]R):
            S is Algebraic_Set of n,R }; then
          consider S be Subset of Funcs(n,[#]R) such that
A20:      o = S & S is Algebraic_Set of n,R;
          thus thesis by A16,A20,ZFMISC_1:25;
        end;
        hence thesis;
      end;
      for n be non zero Nat holds P[n] from NAT_1:sch 10(A14,A1);
      hence thesis;
    end;
