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 Th41:
    for q1,q2 be Ideal of A st q1 in PRIMARY(A,p) & q2 in PRIMARY(A,p)
    holds q1 /\ q2 in PRIMARY(A,p)
    proof
      let q1,q2 be Ideal of A;
      set M = {I where I is primary Ideal of A: I is p-primary };
      assume
A1:   q1 in PRIMARY(A,p) & q2 in PRIMARY(A,p); then
      consider Q1 be primary Ideal of A such that
A2:   Q1 = q1 and
A3:   Q1 is p-primary;
A4:   Q1 <> [#]A;
      consider Q2 be primary Ideal of A such that
A5:   Q2 = q2 and
A6:   Q2 is p-primary by A1;
      set Q3 = Q1 /\ Q2;
A7:   sqrt Q3 = sqrt(Q1) /\ sqrt (Q2) by Th12;
A8:   Q3 <> [#]A by A4, XBOOLE_1:17;
A9:   for x,y be Element of A st x*y in Q3 & not x in Q3 holds y in sqrt Q3
      proof
        let x,y be Element of A;
        assume
A10:    x*y in Q3 & not x in Q3;
        assume not y in sqrt Q3; then
A12:    not (y in sqrt(Q1) /\ sqrt (Q2)) by Th12;
        per cases by A10,XBOOLE_0:def 4;
          suppose
            x*y in Q3 & not(x in Q1); then
            x*y in Q1 & not(x in Q1) by XBOOLE_0:def 4;
            hence contradiction by A3,A6,A12,Def4;
          end;
          suppose
            x*y in Q3 & not(x in Q2); then
            x*y in Q2 & not(x in Q2) by XBOOLE_0:def 4;
            hence contradiction by A3,A6,A12,Def4;
          end;
        end;
        reconsider Q3 as primary Ideal of A by A9,A8,Th33;
        Q3 is p-primary by A3,A6,A7;
        hence thesis by A2,A5;
      end;
