 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 :: 4.1 f) Flip
  for u,w being Element of R,
      x being Subset of R st
  f.u = f.w holds
    u in (Flip ff_0 f).x iff w in (Flip ff_0 f).x
  proof
    let u,w be Element of R,
        x be Subset of R;
    assume
AA: f.u = f.w;
A3: (Flip ff_0 f).x = { w where w is Element of R : f.w c= x } by FlipFF;
    thus u in (Flip ff_0 f).x implies w in (Flip ff_0 f).x
    proof
      assume u in (Flip ff_0 f).x; then
      consider v being Element of R such that
A2:   u = v & f.v c= x by A3;
      thus w in (Flip ff_0 f).x by A3,AA,A2;
    end;
    assume w in (Flip ff_0 f).x; then
    ex v being Element of R st w = v & f.v c= x by A3;
    hence thesis by A3,AA;
  end;
