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

theorem For3A: :: formula (3)
  for w,u being Element of R holds
    [w,u] in the InternalRel of R iff u in (tau R).w
  proof
    let w,u be Element of R;
    thus [w,u] in the InternalRel of R implies u in (tau R).w
    proof
      assume [w,u] in the InternalRel of R; then
      u in Im(the InternalRel of R,w) by RELAT_1:169;
      hence thesis by DefTau;
    end;
    assume u in (tau R).w; then
    u in Im(the InternalRel of R,w) by DefTau; then
    w in Coim(the InternalRel of R,u) by ImCoim; then
    consider x being object such that
S1: [w,x] in the InternalRel of R & x in {u} by RELAT_1:def 14;
    thus thesis by S1,TARSKI:def 1;
  end;
