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

theorem Th6:
  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 Class(EqRel(R,I),a) = 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);
  thus Class(E,a) = Class(E,b) implies a-b in I
  proof
    assume Class(E,a) = Class(E,b);
    then a in Class(E,b) by EQREL_1:23;
    hence thesis by Th5;
  end;
  assume a-b in I;
  then a in Class(E,b) by Th5;
  hence thesis by EQREL_1:23;
end;
