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 p be prime Ideal of A, F be non empty FinSequence of PRIMARY(A,p)
    holds meet rng F in PRIMARY(A,p)
    proof
      let p be prime Ideal of A, F be non empty FinSequence of PRIMARY(A,p);
      set q = meet rng F;
      meet rng F in PRIMARY(A,p)
      proof
:: By Induction of len F
A1:     len F <> 0;
        defpred P[Nat] means for F be non empty FinSequence of PRIMARY(A,p) st
        $1 = len F holds meet rng F in PRIMARY(A,p);
A2:     for n be Nat st n>=1 holds P[n] implies P[n+1]
        proof
          let n be Nat such that
A3:       n>=1 and
A4:       P[n];
          for F be non empty FinSequence of PRIMARY(A,p) st
          n+1 = len F holds meet rng F in PRIMARY(A,p)
          proof
            let F be non empty FinSequence of PRIMARY(A,p);
            assume
A5:         n+1 = len F;
            len F <> 0; then
            consider G be FinSequence of PRIMARY(A,p),
            q be Element of PRIMARY(A,p) such that
A6:         F = G^<*q*> by FINSEQ_2:19;
A7:         Seg(len <*q*>) = dom <*q*> by FINSEQ_1:def 3
            .= Seg 1 by FINSEQ_1:def 8;
A8:         len F = len G + len <*q*> by A6,FINSEQ_1:22
            .= len G + 1 by A7,FINSEQ_1:6;
        reconsider G as non empty FinSequence of PRIMARY(A,p) by A3,A5,A8;
A9:        meet rng G in PRIMARY(A,p) by A4,A5,A8;
            dom G = Seg n by A5,A8,FINSEQ_1:def 3; then
A10:        1 in dom G by A3;
            dom <*q*> = Seg 1 by FINSEQ_1:def 8; then
A12:        rng G <> {} & rng <*q*> <> {} by A10,RELAT_1:42;
            q in {I where I is primary Ideal of A:I is p-primary}; then
            consider q1 be primary Ideal of A such that
A13:        q1 = q & q1 is p-primary;
            consider q2 be primary Ideal of A such that
A14:        q2 = meet rng G & q2 is p-primary by A9;
            q2 /\ q1 in {I where I is primary Ideal of A: I is p-primary}
            by A13,A9,A14,Th41; then
            consider q3 be primary Ideal of A such that
A15:        q3 = q2 /\ q1 and
A16:        q3 is p-primary;
A17:        meet rng F = meet(rng(G) \/ rng <*q*>) by A6,FINSEQ_1:31
            .= meet(rng(G)) /\ meet(rng <*q*>) by A12,SETFAM_1:9
            .= meet(rng(G)) /\ meet {q} by FINSEQ_1:38
            .= q3 by A15, A14,A13,SETFAM_1:10;
            reconsider Q = meet rng F as primary Ideal of A by A17;
            thus thesis by A17,A16;
          end;
          hence thesis;
        end;
A18:    P[1]
        proof
          for F be non empty FinSequence of PRIMARY(A,p) st
          1 = len F holds meet rng F in PRIMARY(A,p)
          proof
            let F be non empty FinSequence of PRIMARY(A,p);
            assume 1 = len F; then
A20:        dom F = {1} by FINSEQ_1:2,def 3; then
A21:        rng F = {F.1} by FUNCT_1:4;
A22:        meet rng F = meet {F.1} by A20,FUNCT_1:4 .= F.1 by SETFAM_1:10;
            F.1 in {F.1} by TARSKI:def 1;
            hence thesis by A22,A21;
          end;
          hence thesis;
        end;
        for n be Nat st n>=1 holds P[n] from NAT_1:sch 8(A18,A2);
        hence thesis by A1,NAT_1:14;
      end;
      hence thesis;
    end;
