
theorem Th54:
  for R being add-associative right_zeroed right_complementable
non empty addLoopStr, I being add-closed non empty Subset of R, a,b,c being
Element of R holds a,b are_congruent_mod I & b,c are_congruent_mod I implies a,
  c are_congruent_mod I
proof
  let R be add-associative right_zeroed right_complementable non empty
  addLoopStr, I be add-closed non empty Subset of R, a,b,c be Element of R;
  assume a,b are_congruent_mod I & b,c are_congruent_mod I;
  then a - b in I & b - c in I;
  then
A1: a - b + (b - c) in I by IDEAL_1:def 1;
  a - b + (b - c) = a + -b + (b - c)
    .= a + (-b + (b - c)) by RLVECT_1:def 3
    .= a + (-b + (b + -c))
    .= a + ((-b + b) + -c) by RLVECT_1:def 3
    .= a + (0.R + -c) by RLVECT_1:5
    .= a + -c by ALGSTR_1:def 2
    .= a - c;
  hence thesis by A1;
end;
