 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;

theorem Th10:
  ex m be maximal Ideal of A st J c= m
    proof
A1: not 1.A in sqrt J by Lm5;
    set S = Ideals(A,J,1.A);
    set P = RelIncl(S);
A2: field P = S by WELLORD2:def 1;
    P partially_orders S by WELLORD2:19,20, 21; then
    consider I being set such that
A3: I is_maximal_in P by A1,A2,Th8, ORDERS_1:63;
    I in S by WELLORD2:def 1,A3; then
    consider p be Subset of A such that
A4: p = I and
A5: p is proper Ideal of A and
A6: J c= p and
    p /\ multClSet(J,1.A) = {};
    for q being Ideal of A st p c= q holds q = p or q is non proper
    proof
      let q be Ideal of A;
      assume
A7:   p c= q;
      per cases;
        suppose
A8:       q is proper;
A9:       multClSet(J,1.A) = {1.A} by Lm6;
A10:      q /\ multClSet(J,1.A) = {}
          proof
            assume q /\ multClSet(J,1.A) <> {}; then
            consider y be object such that
A12:        y in q /\ multClSet(J,1.A) by XBOOLE_0:def 1;
A13:        y in q & y in multClSet(J,1.A) by A12,XBOOLE_0:def 4;
            1.A in q by A9,A13,TARSKI:def 1;
            hence contradiction by A8,IDEAL_1:19;
          end;
          J c= q by A6,A7; then
A14:      q in S by A8,A10;
          [p,q] in P by A2,A3,A4,A7,A14,WELLORD2:def 1;
          hence thesis by A2,A3,A4,A14;
        end;
        suppose q is non proper;
          hence thesis;
        end;
      end; then
      p is quasi-maximal;
      hence thesis by A5,A6;
    end;
