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
    holds x in q implies q % ({x}-Ideal) = [#]A
    proof
      let q be Ideal of A, x be Element of A;
      set I = {x}-Ideal;
      x in q implies (q %I) = [#]A
      proof
        assume
A2:     x in q;
        1.A in (q % I)
        proof
          q = q-Ideal by IDEAL_1:44; then
A4:       I c= q by A2,IDEAL_1:67;
          1.A*I c= q by A4,IDEAL_1:71; then
          1.A in {a where a is Element of A: a*({x}-Ideal) c= q};
          hence thesis by IDEAL_1:def 23;
        end; then
        not (q % I) is proper by IDEAL_1:19;
        hence thesis;
      end;
      hence thesis;
    end;
