 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) g) general version
  for u,w being Element of R,
      x being Subset of R st
    f.u = f.w holds u in (ff_0 f).x iff w in (ff_0 f).x
  proof
    let u,w be Element of R,
        x be Subset of R;
    assume AA: f.u = f.w;
A3: (ff_0 f).x = { w where w is Element of R : f.w meets x }
        by Defff;
    thus u in (ff_0 f).x implies w in (ff_0 f).x
    proof
      assume
A1:   u in (ff_0 f).x;
      consider v being Element of R such that
A2:   u = v & f.v meets x by A1,A3;
      thus w in (ff_0 f).x by A3,AA,A2;
    end;
    assume w in (ff_0 f).x; then
    consider v being Element of R such that
A2: w = v & f.v meets x by A3;
    thus thesis by A3,AA,A2;
  end;
