reserve R for commutative Ring;
reserve A for non degenerated commutative Ring;
reserve I,J,q for Ideal of A;
reserve p for prime Ideal of A;
reserve M,M1,M2 for Ideal of A/q;

theorem
    for a be non empty FinSequence of Ideals(A), p be prime Ideal of A
    holds
    meet rng a c= p implies ex i be object st i in dom a & a.i c= p
    proof
      let a be non empty FinSequence of Ideals(A), p be prime Ideal of A;
      defpred P[Nat] means
      for a be non empty FinSequence of Ideals(A), p be prime Ideal of A
      st len a = $1 holds
      meet rng a c= p implies ex i be object st i in dom a & a.i c= p;
A1:   P[1] by Th4;
A2:   for n be non zero Nat st P[n] holds P[n+1]
      proof
        let n be non zero Nat;
        assume
A3:     P[n];
        for a be non empty FinSequence of Ideals(A), p be prime Ideal of A
        st len a = n+1 holds
        (meet rng a c= p implies ex i be object st i in dom a & a.i c= p)
        proof
          let a be non empty FinSequence of Ideals(A),p be prime Ideal of A;
          assume
A4:       len a = n+1; then
A5:       a = a|n ^<*a/.(n+1)*> by FINSEQ_5:21;
A6:       len (a|n) = n by A4,FINSEQ_1:59,NAT_1:11;
          reconsider an = a|n as non empty FinSequence of Ideals(A);
A7:       dom an = Seg (len an) by FINSEQ_1:def 3 .= Seg n
          by A4,FINSEQ_1:59,NAT_1:11;
A8:       dom a = Seg (n+1) by A4,FINSEQ_1:def 3;
A9:       rng an <> {} & rng <*a/.(n+1)*> <> {} by RELAT_1:41;
A10:      meet rng a = meet (rng an \/ rng <*a/.(n+1)*>) by FINSEQ_1:31,A5
          .= meet(rng an) /\ meet rng(<*a/.(n+1)*>) by A9,SETFAM_1:9;
          meet rng a c= p implies
          ex i be object st i in dom a & a.i c= p
          proof   ::::A
            assume
A11:        meet rng a c= p;
            reconsider I1 = meet(rng an) as Ideal of A by Th3;
            reconsider I2 = meet rng(<*a/.(n+1)*>) as Ideal of A by Th3;
            per cases by Th1,A11,A10;
              suppose I1 c= p; then
                consider i be object such that
A13:            i in dom an & an.i c= p by A6,A3;
A14:            a.i c= p by A13, A5,FINSEQ_1:def 7;
A15:            Seg n c= Seg (n+1) by FINSEQ_1:5,NAT_1:11;
                consider i1 be object such that
A16:            i1 = i and
A17:            i1 in dom a & a.i c= p by A14,A13,A7,A8,A15;
                thus thesis by A16,A17;
              end;
              suppose
A18:            I2 c= p;
                rng <*a/.(n+1)*> = {a/.(n+1)} by FINSEQ_1:39; then
A19:            meet rng <*a/.(n+1)*> = a/.(n+1) by SETFAM_1:10
                .= a.(n+1) by A8,FINSEQ_1:4,PARTFUN1:def 6;
                consider i be object such that
                i = n+1 and
A20:            i in dom a & a.i c= p by A18,A19,A8,FINSEQ_1:4;
                thus thesis by A20;
              end;
            end;
            hence thesis;
          end;
          hence thesis;
        end;
A21:    for i being non zero Nat holds P[i] from NAT_1:sch 10(A1,A2);
        reconsider n = len a as non zero Nat;
        n = len a;
        hence thesis by A21;
      end;
