
theorem Th53:
  for R being add-associative right_zeroed right_complementable
well-unital right-distributive non empty doubleLoopStr, I being right-ideal
  non empty Subset of R, a,b being Element of R holds a,b are_congruent_mod I
  implies b,a are_congruent_mod I
proof
  let R be add-associative right_zeroed right_complementable
right-distributive well-unital non empty doubleLoopStr, I be right-ideal non
  empty Subset of R, a,b be Element of R;
  assume a,b are_congruent_mod I;
  then a - b in I;
  then
A1: -(a - b) in I by IDEAL_1:14;
  b - a - (-(a - b)) = b - a + -(-(a - b))
    .= b - a + (a - b) by RLVECT_1:17
    .= b + -a + (a - b)
    .= b + (-a + (a - b)) by RLVECT_1:def 3
    .= b + (-a + (a + -b))
    .= b + ((-a + a) + -b) by RLVECT_1:def 3
    .= b + (0.R + -b) by RLVECT_1:5
    .= b + -b by ALGSTR_1:def 2
    .= 0.R by RLVECT_1:5;
  then b - a = -(a - b) by RLVECT_1:21;
  hence thesis by A1;
end;
