
theorem
  for R being Abelian add-associative right_zeroed right_complementable
  well-unital distributive associative non trivial doubleLoopStr, I being
  add-closed non empty Subset of R, a,b,c,d being Element of R holds a,b
are_congruent_mod I & c,d are_congruent_mod I implies a+c,b+d are_congruent_mod
  I
proof
  let R be Abelian add-associative right_zeroed right_complementable
  well-unital distributive associative non trivial doubleLoopStr, I be
  add-closed non empty Subset of R, a,b,c,d be Element of R;
  assume a,b are_congruent_mod I & c,d are_congruent_mod I;
  then a - b in I & c - d in I;
  then
A1: (a - b) + (c - d) in I by IDEAL_1:def 1;
  (a + c) - (b + d) = (a + c) + (-(b + d))
    .= (a + c) + (-d + -b) by RLVECT_1:31
    .= a + (c + (-d + -b)) by RLVECT_1:def 3
    .= a + ((c + -d) + -b) by RLVECT_1:def 3
    .= (a + -b) + (c + -d) by RLVECT_1:def 3
    .= (a - b) + (c + -d)
    .= (a - b) + (c - d);
  hence thesis by A1;
end;
