 reserve f,g for Function;
 reserve R for non empty reflexive RelStr;
 reserve R for non empty RelStr;
 reserve f for Function of the carrier of R, bool the carrier of R;

theorem Propj: :: j)
  for f being Function of the carrier of R, bool the carrier of R
  for x,y being Subset of R holds
    (ff_0 f).(x \/ y) = (ff_0 f).x \/ (ff_0 f).y
  proof
    let f be Function of the carrier of R, bool the carrier of R;
    let x,y be Subset of R;
AA: (ff_0 f).(x \/ y) =
      { u where u is Element of R : f.u meets (x \/ y) } by Defff;
AB: (ff_0 f).x =
      { u where u is Element of R : f.u meets x } by Defff;
AC: (ff_0 f).y =
      { u where u is Element of R : f.u meets y } by Defff;
    thus (ff_0 f).(x \/ y) c= (ff_0 f).x \/ (ff_0 f).y
    proof
      let t be object;
      assume t in (ff_0 f).(x \/ y); then
      consider u being Element of R such that
A1:   t = u & f.u meets (x \/ y) by AA;
      f.u meets x or f.u meets y by A1,XBOOLE_1:70; then
      t in (ff_0 f).x or t in (ff_0 f).y by A1,AB,AC;
      hence thesis by XBOOLE_0:def 3;
    end;
    let t be object;
    assume t in (ff_0 f).x \/ (ff_0 f).y; then
    per cases by XBOOLE_0:def 3;
    suppose t in (ff_0 f).x; then
      consider u being Element of R such that
A1:     t = u & f.u meets x by AB;
      f.u meets (x \/ y) by XBOOLE_1:70,A1;
      hence thesis by A1,AA;
    end;
    suppose t in (ff_0 f).y; then
      consider u being Element of R such that
A1:     t = u & f.u meets y by AC;
      f.u meets (x \/ y) by XBOOLE_1:70,A1;
      hence thesis by A1,AA;
    end;
  end;
