
theorem
  for R being add-associative right_zeroed right_complementable
  commutative distributive non empty doubleLoopStr, I being add-closed
  right-ideal 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 add-associative right_zeroed right_complementable commutative
distributive non empty doubleLoopStr, I be add-closed right-ideal non empty
  Subset of R, a,b,c,d be Element of R;
  assume that
A1: a,b are_congruent_mod I and
A2: c,d are_congruent_mod I;
  c - d in I by A2;
  then
A3: (c - d) * b in I by IDEAL_1:def 3;
A4: (c - d) * b = (c + -d) * b
    .= c * b + (-d) * b by VECTSP_1:def 3;
  (a - b) * c = (a + -b) * c
    .= a * c + (-b) * c by VECTSP_1:def 3;
  then
A5: ((a - b) * c) + ((c - d) * b) = a * c + ((-b) * c + (c * b + (-d) * b))
  by A4,RLVECT_1:def 3
    .= a * c + (((-b) * c + c * b) + (-d) * b) by RLVECT_1:def 3
    .= a * c + ((-(b * c) + c * b) + (-d) * b) by VECTSP_1:9
    .= a * c + (0.R + (-d) * b) by RLVECT_1:5
    .= a * c + (-d) * b by ALGSTR_1:def 2
    .= a * c + -(d * b) by VECTSP_1:9
    .= a * c - b * d;
  a - b in I by A1;
  then (a - b) * c in I by IDEAL_1:def 3;
  then ((a - b) * c) + ((c - d) * b) in I by A3,IDEAL_1:def 1;
  hence thesis by A5;
end;
