reserve R for Ring,
  I for Ideal of R,
  a, b for Element of R;

theorem Th9:
  Class(EqRel(R,{0.R}),a) = {a}
proof
  set E = EqRel(R,{0.R});
  thus Class(E,a) c= {a}
  proof
    let A be object;
    assume
A1: A in Class(E,a);
    then reconsider A as Element of R;
    [A,a] in E by A1,EQREL_1:19;
    then A-a in {0.R} by Def5;
    then A-a = 0.R by TARSKI:def 1;
    then A = a by RLVECT_1:21;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {a};
  then
A2: x = a by TARSKI:def 1;
  a-a = 0.R & 0.R in {0.R} by RLVECT_1:15,TARSKI:def 1;
  then [x,a] in E by A2,Def5;
  hence thesis by EQREL_1:19;
end;
