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 q be Ideal of A, x be Element of A st q in PRIMARY(A,p)
    holds not(x in q) implies q % ({x}-Ideal) in PRIMARY(A,p)
    proof
      let q be Ideal of A, x be Element of A;
      assume
A1:   q in PRIMARY(A,p);
      set I = {x}-Ideal;
      consider q1 be primary Ideal of A such that
A2:   q1 = q and
A3:   q1 is p-primary by A1;
      set M = {a where a is Element of A: a*I c= q};
      reconsider J = (q % I) as Ideal of A;
      not(x in q) implies (q % I) in PRIMARY(A,p)
      proof
        assume
A4:     not x in q;
        not 1.A in (q % I)
        proof
          assume 1.A in (q % I); then
          1.A in M by IDEAL_1:def 23; then
          consider y1 being Element of A such that
A6:       1.A = y1 and
A7:       y1*I c= q;
          I c= q by A7,A6,IDEAL_1:71;
          hence contradiction by A4,IDEAL_1:66;
        end; then
A9:     (q % I) is proper;
        for y be Element of A holds y in (q % I) implies y in p
        proof
          let y be Element of A;
          assume y in (q % I); then
          y in M by IDEAL_1:def 23; then
          consider y1 being Element of A such that
A11:      y1= y and
A12:      y1*I c= q;
          x in I by IDEAL_1:66; then
A14:      y1*x in {y1*i where i is Element of A : i in I};
          y1*x in y1*I by A14,IDEAL_1:def 18;
          hence thesis by A11, A3, Th33, A4, A2, A12;
        end; then
A16:    q % I c= p;
A17:    p = sqrt (q % I) by A16,TOPZARI1:25, A3,A2,A9,TOPZARI1:21,IDEAL_1:85;
:::::::::::: to prove (q % I) is primary
        set Q = (q % I);
        reconsider Q as proper Ideal of A by A9;
A18:    for a,b be Element of A st a*b in Q & not b in p holds a in Q
        proof
          let a,b be Element of A;
          assume
A19:      a*b in Q & not b in p; then
          a*b in M by IDEAL_1:def 23; then
          consider ab1 being Element of A such that
A20:      ab1 = a*b and
A21:      ab1*I c= q1 by A2;
A22:      for i be Element of A st i in I holds a*i in q1
          proof
            let i be Element of A;
            assume i in I; then
            ab1*i in {ab1*i1 where i1 is Element of A : i1 in I}; then
A24:        ab1*i in ab1*I by IDEAL_1:def 18;
            (a*b)*i = (i*a)*b by GROUP_1:def 3;
            hence thesis by A3,Def4,A20,A21,A24,A19;
          end;
          a*I c= q
          proof
            for z be object holds z in a*I implies z in q1
            proof
              let z be object;
              assume z in a*I; then
              z in {a*i1 where i1 is Element of A : i1 in I}
              by IDEAL_1:def 18; then
              consider i0 be Element of A such that
A27:          z = a*i0 and
A28:          i0 in I;
              thus thesis by A27,A28,A22;
            end;
            hence thesis by A2;
          end; then
          a in {y where y is Element of A: y*I c= q};
          hence thesis by IDEAL_1:def 23;
        end;
        reconsider Q as primary Ideal of A by A18,A17,Th45;
        Q is p-primary by A17;
        hence thesis;
      end;
      hence thesis;
    end;
