reserve R for Ring,
  I for Ideal of R,
  a, b for Element of R;
reserve R for add-associative right_zeroed right_complementable Abelian
    distributive left_unital non empty doubleLoopStr;
reserve I for Ideal of R;
reserve a,b for Element of R;
reserve x, y for Element of R/I;

theorem Th16:
  for R being Ring, I being Ideal of R
  holds I is proper iff R/I is non degenerated
proof
  let R be Ring, I be Ideal of R;
  set E = EqRel(R,I);
A1: 1.R-0.R = 1.R by RLVECT_1:13;
A2: 0.(R/I) = Class(E,0.R) & 1.(R/I) = Class(E,1.R) by Def6;
  thus I is proper implies R/I is non degenerated
  by A2,Th6,A1,IDEAL_1:19;
  assume
A3: R/I is non degenerated;
  assume not I is proper;
  then 1.R in I by IDEAL_1:19;
  hence thesis by A2,A1,A3,Th6;
end;
