
theorem
  for R being left_zeroed right_zeroed right_add-cancelable
right-distributive non empty doubleLoopStr, I being add-closed Subset of R, J
,K being right-ideal non empty Subset of R holds I % (J + K) = (I % J) /\ (I
  % K)
proof
  let R be left_zeroed right_zeroed right_add-cancelable right-distributive
non empty doubleLoopStr, I be add-closed Subset of R, J,K be right-ideal non
  empty Subset of R;
A1: now
    let u be object;
    assume u in I % (J + K);
    then consider a being Element of R such that
A2: u = a and
A3: a*(J+K) c= I;
    now
      let u be object;
      assume u in a*J;
      then
A4:   ex j being Element of R st u = a*j & j in J;
      J c= J+K by Th73;
      then u in {a*j9 where j9 is Element of R : j9 in J+K} by A4;
      hence u in I by A3;
    end;
    then a*J c= I;
    then
A5: u in (I % J) by A2;
    now
      let u be object;
      assume u in a*K;
      then
A6:   ex j being Element of R st u = a*j & j in K;
      K c= J+K by Th74;
      then u in {a*j9 where j9 is Element of R : j9 in J+K} by A6;
      hence u in I by A3;
    end;
    then a*K c= I;
    then u in (I % K) by A2;
    hence u in (I % J) /\ (I % K) by A5;
  end;
  now
    let u be object;
    assume u in (I % J) /\ (I % K);
    then
A7: ex x being Element of R st u = x & x in (I % J) & x in (I % K);
    then consider a being Element of R such that
A8: u = a and
A9: a*J c= I;
    consider b being Element of R such that
A10: u = b and
A11: b*K c= I by A7;
    now
      let v be object;
      assume v in a*(J+K);
      then consider j being Element of R such that
A12:  v = a*j and
A13:  j in J+K;
      consider x9,y being Element of R such that
A14:  j = x9 + y and
A15:  x9 in J & y in K by A13;
A16:  a*x9 in a*J & b*y in {b*j9 where j9 is Element of R : j9 in K} by A15;
      v = a*x9 + b*y by A8,A10,A12,A14,VECTSP_1:def 2;
      hence v in I by A9,A11,A16,Def1;
    end;
    then a*(J+K) c= I;
    hence u in I % (J + K) by A8;
  end;
  hence thesis by A1,TARSKI:2;
end;
