 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)
  for u,w being Element of R,
      x being Subset of R st
  (tau R).u = (tau R).w holds
    u in (f_0 R).x iff w in (f_0 R).x
  proof
    let u,w be Element of R,
        x be Subset of R;
    assume
AA: (tau R).u = (tau R).w;
A3: (f_0 R).x = { w where w is Element of R : (tau R).w meets x } by Defff;
    thus u in (f_0 R).x implies w in (f_0 R).x
    proof
      assume u in (f_0 R).x; then
      consider v being Element of R such that
A2:   u = v & (tau R).v meets x by A3;
      thus w in (f_0 R).x by A3,AA,A2;
    end;
    assume w in (f_0 R).x; then
A2: ex v being Element of R st
    w = v & (tau R).v meets x by A3;
    thus thesis by A3,AA,A2;
  end;
