
theorem
  for R being Abelian left_zeroed right_zeroed right_complementable
  left_unital add-associative left-distributive non empty doubleLoopStr, I
  being add-closed left-ideal non empty Subset of R, J being Subset of R, K
  being non empty Subset of R st J c= I holds I /\ (J + K) = J + (I /\ K)
proof
  let R be Abelian left_zeroed right_zeroed right_complementable left_unital
  add-associative left-distributive non empty doubleLoopStr, I be add-closed
left-ideal non empty Subset of R, J be Subset of R, K be non empty Subset of
  R;
  assume
A1: J c= I;
A2: now
    let u be object;
    assume u in J + (I /\ K);
    then consider j,ik being Element of R such that
A3: u = j + ik and
A4: j in J and
A5: ik in (I /\ K);
A6: ex z being Element of R st z = ik & z in I & z in K by A5;
    then reconsider k9 = ik as Element of K;
    u = j + k9 by A3;
    then
A7: u in J + K by A4;
    reconsider j9 = j as Element of I by A1,A4;
    reconsider i9 = ik as Element of I by A6;
    u = j9 + i9 by A3;
    then u in I by Def1;
    hence u in I /\ (J + K) by A7;
  end;
  now
    let u be object;
    assume u in I /\ (J + K);
    then consider v being Element of R such that
A8: u = v and
A9: v in I and
A10: v in J + K;
    consider j,k being Element of R such that
A11: v = j + k and
A12: j in J and
A13: k in K by A10;
    reconsider j9 = j as Element of I by A1,A12;
    -j9 in I by Th13;
    then (j9 + k) + -j9 in I by A9,A11,Def1;
    then (j9 + -j9) + k in I by RLVECT_1:def 3;
    then 0.R + k in I by RLVECT_1:5;
    then k in I by ALGSTR_1:def 2;
    then k in (I /\ K) by A13;
    hence u in J + (I /\ K) by A8,A11,A12;
  end;
  hence thesis by A2,TARSKI:2;
end;
