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;

theorem Th4:
  A is domRing implies {0.A} = ZeroDiv_Set(A)
  proof
    assume
A0: A is domRing;
    0.A is Zero_Divisor of A by Th1; then
A1: 0.A in ZeroDiv_Set(A);
A2: {0.A} is Subset of ZeroDiv_Set(A) by A1,SUBSET_1:33;
    for o st o in ZeroDiv_Set(A) holds o in {0.A}
    proof
      let o;
      assume o in ZeroDiv_Set(A); then
      consider a be Element of [#]A such that
A4:   o = a and
A5:   a is Zero_Divisor of A;
      consider b be Element of A such that
A6:   b <> 0.A and
A7:   a*b = 0.A by A5,Def1;
      a = 0.A by A6,A0,A7,VECTSP_2:def 1;
      hence thesis by A4,TARSKI:def 1;
    end; then
    ZeroDiv_Set(A) c= {0.A};
    hence thesis by A2,XBOOLE_0:def 10;
  end;
