reserve R,R1 for commutative Ring;
reserve A,B for non degenerated commutative Ring;
reserve o,o1,o2 for object;
reserve r,r1,r2 for Element of R;
reserve a,a1,a2,b,b1 for Element of A;
reserve f for Function of R, R1;
reserve p for Element of Spectrum A;
reserve S for non empty multiplicatively-closed Subset of R;
reserve u,v,w,x,y,z for Element of Frac(S);
reserve a, b, c for Element of Frac(S);
reserve x, y, z for Element of S~R;
reserve S for without_zero non empty multiplicatively-closed Subset of A;
reserve p for Element of Spectrum A;
reserve a,m,n for Element of A~p;
reserve f for Function of A,B;
reserve x for object;

theorem
   A is Field implies Ideals(A) = {{0.A}, the carrier of A}
   proof
     assume
A1:  A is Field;
A2:  x in Ideals(A) implies x in {{0.A}, the carrier of A}
     proof
       assume x in Ideals(A); then
       x in the set of all I where I is Ideal of A by RING_2:def 3; then
       consider x1 being Ideal of A such that
A4:    x1 = x;
       x = {0.A} or x = the carrier of A by A1,A4,IDEAL_1:22;
       hence thesis by TARSKI:def 2;
     end;
     x in {{0.A}, the carrier of A} implies x in Ideals(A)
     proof
       assume x in {{0.A}, the carrier of A}; then
       per cases by TARSKI:def 2;
         suppose
           x = {0.A}; then
           x in the set of all I where I is Ideal of A;
           hence thesis by RING_2:def 3;
         end;
         suppose x = the carrier of A; then
           x is Ideal of A by IDEAL_1:10; then
           x in the set of all I where I is Ideal of A;
           hence thesis by RING_2:def 3;
         end;
       end;
      hence thesis by A2,TARSKI:2;
    end;
