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 Th6:
    for I be proper Ideal of A, F be non empty FinSequence of Ideals(A) holds
    rng((%I)*F) <> {} & rng F <> {} &
    meet rng((%I)*F) c= the carrier of A
    proof
      let I be proper Ideal of A, F be non empty FinSequence of Ideals(A);
      reconsider J = meet rng F as Ideal of A by Th3;
A1:   rng F c= bool the carrier of A by XBOOLE_1:1; then
      reconsider F1 = F as non empty FinSequence of bool the carrier of A
        by FINSEQ_1:def 4;
A2:   dom %I = bool the carrier of A by FUNCT_2:def 1;
A3:   1 in dom F by FINSEQ_5:6;
      1 in dom (%I*F) by A1,A2,RELAT_1:27,A3;
      hence thesis by A3,FUNCT_1:3;
    end;
