 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 Propm: :: m)
  for f being Function of the carrier of R, bool the carrier of R
  for x,y being Subset of R holds
    (Flip ff_0 f).x /\ (Flip ff_0 f).y = (Flip ff_0 f).(x /\ 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 (Flip ff_0 f).x /\ (Flip ff_0 f).y c= (Flip ff_0 f).(x /\ y)
    proof
      let t be object;
      assume zz: t in (Flip ff_0 f).x /\ (Flip ff_0 f).y; then
ZZ:   t in (Flip ff_0 f).x & t in (Flip ff_0 f).y by XBOOLE_0:def 4; then
      t in ((ff_0 f).x`)` by ROUGHS_2:def 14; then
Z1:   not t in ((ff_0 f).x`) by XBOOLE_0:def 5;
      t in ((ff_0 f).y`)` by ZZ,ROUGHS_2:def 14; then
V1:   not t in ((ff_0 f).y`) by XBOOLE_0:def 5;
      not t in (ff_0 f).(x /\ y)`
      proof
        assume t in (ff_0 f).(x /\ y)`; then
        consider w being Element of R such that
A1:     t = w & f.w meets (x /\ y)` by AA;
        f.w meets (x` \/ y`) by XBOOLE_1:54,A1; then
        per cases by XBOOLE_1:70;
        suppose f.w meets x`;
          hence thesis by Z1,A1,AB;
        end;
        suppose f.w meets y`;
          hence thesis by V1,AC,A1;
        end;
      end; then
      t in ((ff_0 f).(x /\ y)`)` by zz,XBOOLE_0:def 5;
      hence thesis by ROUGHS_2:def 14;
    end;
    let t be object;
    assume v0: t in (Flip ff_0 f).(x /\ y); then
    t in ((ff_0 f).(x /\ y)`)` by ROUGHS_2:def 14; then
ww: not t in { u where u is Element of R : f.u meets (x /\ y)` }
      by AA,XBOOLE_0:def 5;
vc: (x /\ y)` = x` \/ y` by XBOOLE_1:54;
w1: not t in (ff_0 f).x`
    proof
      assume t in (ff_0 f).x`; then
      consider w being Element of R such that
A1:   t = w & f.w meets x` by AB;
      f.w meets (x /\ y)` by vc,A1,XBOOLE_1:63,XBOOLE_1:7;
      hence thesis by ww,A1;
    end;
z1: not t in (ff_0 f).y`
    proof
      assume t in (ff_0 f).y`; then
      consider w being Element of R such that
A1:   t = w & f.w meets y` by AC;
      f.w meets (x /\ y)` by vc,A1,XBOOLE_1:63,7;
      hence thesis by ww,A1;
    end;
    t in ((ff_0 f).x`)` by w1,v0,XBOOLE_0:def 5; then
W1: t in (Flip ff_0 f).x by ROUGHS_2:def 14;
    t in ((ff_0 f).y`)` by z1,v0,XBOOLE_0:def 5; then
    t in (Flip ff_0 f).y by ROUGHS_2:def 14;
    hence thesis by W1,XBOOLE_0:def 4;
  end;
