 reserve f,g for Function;
 reserve R for non empty reflexive RelStr;
 reserve R for non empty RelStr;

theorem UApF0:
  f_0 R = UAp R
  proof
    set ff = f_0 R;
    set gg = UAp R;
    for x being Subset of R holds ff.x = gg.x
    proof
      let x be Subset of R;
WW:   { u where u is Element of R : (tau R).u meets x } =
            { w where w is Element of R :
               Class (the InternalRel of R, w) meets x }
      proof
        thus { u where u is Element of R : (tau R).u meets x } c=
            { w where w is Element of R :
            Class (the InternalRel of R, w) meets x }
        proof
          let t be object;
          assume t in { u where u is Element of R :
             (tau R).u meets x }; then
          consider u being Element of R such that
W1:       u = t & (tau R).u meets x;
          consider tt being object such that
W2:       tt in (tau R).u & tt in x by XBOOLE_0:3,W1;
W4:       (tau R).u = Im (the InternalRel of R, u) by DefTau;
          reconsider ttt = tt as Element of R by W2;
          thus thesis by W1,W4;
        end;
        let t be object;
        assume t in { w where w is Element of R :
            Class (the InternalRel of R, w) meets x }; then
        consider tt being Element of R such that
H1:     tt = t & Class (the InternalRel of R, tt) meets x;
        consider wx being object such that
H2:     wx in Class (the InternalRel of R, tt) & wx in x by H1,XBOOLE_0:3;
        reconsider wxx = wx as Element of R by H2;
        (tau R).tt = Im (the InternalRel of R, tt) by DefTau;
        hence thesis by H1;
      end;
      ff.x = UAp x by WW,Defff
          .= gg.x by ROUGHS_2:def 11;
      hence thesis;
    end;
    hence thesis;
  end;
