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

theorem
  Class EqRel(R,[#]R) = {the carrier of R}
proof
  set E = EqRel(R,[#]R);
  thus Class E c= {the carrier of R}
  proof
    let A be object;
    assume A in Class E;
    then consider x being object such that
A1: x in the carrier of R and
A2: A = Class(E,x) by EQREL_1:def 3;
    reconsider x as Element of R by A1;
    Class(E,x) = the carrier of R
    proof
      thus Class(E,x) c= the carrier of R;
      let a be object;
      assume a in the carrier of R;
      then reconsider a as Element of R;
      a-x in [#]R;
      then [a,x] in E by Def5;
      hence thesis by EQREL_1:19;
    end;
    hence thesis by A2,TARSKI:def 1;
  end;
  let A be object;
  assume A in {the carrier of R};
  then A = the carrier of R by TARSKI:def 1
    .= Class(E,0.R) by Th7;
  hence thesis by EQREL_1:def 3;
end;
