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

theorem UncEqTau:
  the InternalRel of R is symmetric implies
    UncertaintyMap R = tau R
  proof
    assume
AA: the InternalRel of R is symmetric;
    set f = UncertaintyMap R,
        g = tau R;
    for x being Element of R holds f.x = g.x
    proof
      let x be Element of R;
Z2:   f.x = Coim(the InternalRel of R,x) by DefUnc;
ZZ:   Im(the InternalRel of R,x) c= Coim(the InternalRel of R,x)
      proof
        let y be object;
        assume y in Im(the InternalRel of R,x); then
        [x,y] in the InternalRel of R by RELAT_1:169; then
B2:     [y,x] in the InternalRel of R by AA,PREFER_1:20;
        x in {x} by TARSKI:def 1;
        hence thesis by B2,RELAT_1:def 14;
      end;
      Coim(the InternalRel of R,x) c= Im(the InternalRel of R,x)
      proof
        let y be object;
        assume y in Coim(the InternalRel of R,x); then
        consider yy being object such that
B2:     [y,yy] in the InternalRel of R & yy in {x} by RELAT_1:def 14;
        yy = x by TARSKI:def 1,B2; then
        [x,y] in the InternalRel of R by B2,PREFER_1:20,AA;
        hence thesis by RELAT_1:169;
      end;
      hence thesis by ZZ,DefTau,Z2;
    end;
    hence thesis;
  end;
