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

theorem Th5:
  for R being add-associative right_zeroed right_complementable
    left-distributive left_unital non empty doubleLoopStr
  for I being Ideal of R, a,b being Element of R
  holds a in Class(EqRel(R,I),b) iff a-b in I
proof
  let R be add-associative right_zeroed right_complementable
    left-distributive left_unital non empty doubleLoopStr;
  let I be Ideal of R, a,b be Element of R;
  set E = EqRel(R,I);
  hereby
    assume a in Class(E,b);
    then [a,b] in E by EQREL_1:19;
    hence a-b in I by Def5;
  end;
  assume a-b in I;
  then [a,b] in E by Def5;
  hence thesis by EQREL_1:19;
end;
