 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 FlipFF:
  for f being Function of the carrier of R, bool the carrier of R
  for x being Subset of R holds
    (Flip ff_0 f).x = { w where w is Element of R : f.w c= x }
  proof
    let f be Function of the carrier of R, bool the carrier of R;
    let x be Subset of R;
ZZ: (ff_0 f).(x`) = { w where w is Element of R : f.w meets x` }
        by Defff;
    thus (Flip ff_0 f).x c= { w where w is Element of R : f.w c= x }
    proof
      let y be object;
      assume
S1:   y in (Flip ff_0 f).x; then
      y in ((ff_0 f).(x`))` by ROUGHS_2:def 14; then
Z1:   not y in (ff_0 f).(x`) by XBOOLE_0:def 5;
      reconsider yy = y as Element of R by S1;
      f.yy misses x` by Z1,ZZ; then
      f.yy c= x by SUBSET_1:24;
      hence thesis;
    end;
    let y be object;
    assume y in { w where w is Element of R : f.w c= x }; then
    consider w being Element of R such that
L1: y = w & f.w c= x;
    reconsider yy = y as Element of R by L1;
    not yy in ((ff_0 f).x`)
    proof
      assume yy in (ff_0 f).x`; then
      consider v being Element of R such that
L2:   yy = v & f.v meets x` by ZZ;
      thus thesis by L1,L2,SUBSET_1:24;
    end; then
    yy in ((ff_0 f).(x`))` by XBOOLE_0:def 5;
    hence thesis by ROUGHS_2:def 14;
  end;
