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

theorem :: (6)
  the InternalRel of R is symmetric iff
    for u,v being Element of R holds
      u in (tau R).v implies v in (tau R).u
  proof
    hereby assume
A1: the InternalRel of R is symmetric;
    let u,v be Element of R;
    assume u in (tau R).v; then
    u in (UncertaintyMap R).v by A1,UncEqTau; then
    [u,v] in the InternalRel of R by For3;
    hence v in (tau R).u by For3A;
    end;
    assume
Z0: for u,v being Element of R holds
      u in (tau R).v implies v in (tau R).u;
    for a,b being object st [a,b] in the InternalRel of R holds
      [b,a] in the InternalRel of R
    proof
      let a,b be object;
      assume
Z1:   [a,b] in the InternalRel of R; then
      reconsider aa = a, bb = b as Element of R by Lemacik;
      bb in (tau R).aa by Z1,For3A; then
      aa in (tau R).bb by Z0;
      hence thesis by For3A;
    end;
    hence thesis by PREFER_1:20;
  end;
