
theorem
  for R being non empty multLoopStr, I,J,K being Subset of R holds (J /\
  K) % I = (J % I) /\ (K % I)
proof
  let R be non empty multLoopStr, I,J,K be Subset of R;
A1: now
    let u be object;
    assume u in (J /\ K) % I;
    then consider a being Element of R such that
A2: u = a and
A3: a*I c= (J /\ K);
    now
      let v be object;
      assume v in a*I;
      then v in J /\ K by A3;
      then ex x being Element of R st v = x & x in J & x in K;
      hence v in K;
    end;
    then a*I c= K;
    then
A4: u in (K % I) by A2;
    now
      let v be object;
      assume v in a*I;
      then v in J /\ K by A3;
      then ex x being Element of R st v = x & x in J & x in K;
      hence v in J;
    end;
    then a*I c= J;
    then u in (J % I) by A2;
    hence u in (J % I) /\ (K % I) by A4;
  end;
  now
    let u be object;
    assume u in (J % I) /\ (K % I);
    then
A5: ex x being Element of R st x = u & x in (J % I) & x in (K % I);
    then consider a being Element of R such that
A6: u = a and
A7: a*I c= J;
    ex b being Element of R st u = b & b*I c= K by A5;
    then for v be object st v in a*I holds v in J /\ K by A6,A7;
    then a*I c= J /\ K;
    hence u in (J /\ K) % I by A6;
  end;
  hence thesis by A1,TARSKI:2;
end;
