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 p implies q % ({x}-Ideal) = q
    proof
      let q be Ideal of A, x be Element of A;
      assume
A1:   q in PRIMARY(A,p);
      set I = {x}-Ideal;
      set M = {a where a is Element of A: a*I c= q};
      consider Q be primary Ideal of A such that
A2:   Q = q and
A3:   Q is p-primary by A1;
      not(x in p) implies (q % I) = q
      proof
        assume
A4:     not x in p;
A5:     for o be object holds o in (q % I) implies o in q
        proof
          let o be object;
          assume o in q % I; then
          o in M by IDEAL_1:def 23; then
          consider o1 being Element of A such that
A7:       o1 = o and
A8:       o1*I c= q;
          x in I by IDEAL_1:66; then
          o1*x in {o1*i1 where i1 is Element of A : i1 in I}; then
          o1*x in o1*I by IDEAL_1:def 18;
          hence thesis by A7,Def4,A2,A3,A4,A8;
        end;
        q % I c= q by A5;
        hence thesis by IDEAL_1:85;
      end;
      hence thesis;
    end;
