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 I be proper Ideal of A, F be non empty FinSequence of Ideals(A) holds
    (%I).(meet rng F) = meet rng((%I)*F)
    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:   dom(%I*F) = dom F by A1,A2,RELAT_1:27;
A4:   rng((%I)*F) <> {} by Th6;
A5:   rng F c= Ideals(A);
A6:   for x be object holds x in (%I).(meet rng F) implies
      x in meet rng((%I)*F)
      proof
        let x be object;
        assume x in (%I).(meet rng F); then
A8:     x in (J%I) by Def1;
        x in {a where a is Element of A: a*I c= J} by A8,IDEAL_1:def 23; then
        consider a be Element of A such that
A9:     a = x & a*I c= J;
        x in meet rng((%I)*F)
        proof
          assume not x in meet rng((%I)*F); then
          consider Y be set such that
A11:      Y in rng((%I)*F) & not x in Y by A4,SETFAM_1:def 1;
          consider i be object such that
A12:      i in dom ((%I)*F) & Y = ((%I)*F).i by A11,FUNCT_1:def 3;
A13:      Y = (%I).(F1.i) by A12,FINSEQ_3:120;
          reconsider i1 = i as Nat by A12;
          i in dom F by A1,A2,RELAT_1:27,A12; then
A14:      F.i in rng F by FUNCT_1:def 3;
          F.i in Ideals(A) by A14; then
          F.i in the set of all I where I is Ideal of A by RING_2:def 3; then
          consider Fi be Ideal of A such that
A15:      Fi = F.i;
A16:      Y = Fi % I by Def1,A13,A15;
          meet rng F c= F.i  by A14,SETFAM_1:3; then
          a*I c= Fi by A9,A15; then
          a in {b where b is Element of A: b*I c= Fi};
          hence contradiction by A11,A9,A16,IDEAL_1:def 23;
        end;
        hence thesis;
      end;
      meet rng((%I)*F) c= (%I).(meet rng F)
      proof
        let x be object;
        assume
A18:    x in meet rng((%I)*F); then
        consider a be Element of A such that
A19:    x = a;
A20:    dom((%I)*F1) = dom F1 & len ((%I)*F1) = len F1 &
        for i be Nat st i in dom((%I)*F1) holds ((%I)*F1).i = (%I).(F1.i)
          by FINSEQ_3:120;
A21:    F is Function of dom F, rng F by FUNCT_2:1;
        for Y be set st Y in rng F holds a*I c= Y
        proof
          let Y be set;
          assume Y in rng F; then
          consider j be object such that
A23:      j in dom F & Y = F.j by A21,FUNCT_2:11;
          F.j in Ideals(A) by A23,A5,FUNCT_1:3;then
          F.j in the set of all I where I is Ideal of A by RING_2:def 3; then
          consider Fj be Ideal of A such that
A24:      Fj = F.j;
A25:      ((%I)*F1).j in rng((%I)*F) by A23,A3,FUNCT_1:def 3;
          x in ((%I)*F).j by A25,A18,SETFAM_1:def 1; then
          x in (%I).(Fj) by A20,A23, A24; then
A26:      x in (Fj%I) by Def1;
          a in {b where b is Element of A: b*I c=Fj} by A19,A26,IDEAL_1:def 23;
          then
          consider b1 be Element of A such that
A27:      a = b1 & b1 * I c= Fj;
          thus thesis by A23,A24,A27;
        end; then
A28:    a*I c= meet rng F by Th6,SETFAM_1:5;
        x in (%I).(meet rng F)
        proof
          assume
A29:      not x in (%I).(meet rng F);
A30:      (%I).(meet rng F) = J%I by Def1;
          not(x in {b where b is Element of A: b*I c= J})
            by A29,A30,IDEAL_1:def 23;
          hence contradiction by A28,A19;
        end;
        hence thesis;
      end;
      hence thesis by A6,TARSKI:2;
    end;
