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

theorem For3: :: property (3)
  for w,u being Element of R holds
    [w,u] in the InternalRel of R iff w in (UncertaintyMap R).u
  proof
    let w,u be Element of R;
    thus [w,u] in the InternalRel of R implies w in (UncertaintyMap R).u
    proof
      assume S1: [w,u] in the InternalRel of R;
      u in {u} by TARSKI:def 1; then
      w in Coim(the InternalRel of R,u) by S1,RELAT_1:def 14;
      hence thesis by DefUnc;
    end;
    assume w in (UncertaintyMap R).u; then
    w in Coim(the InternalRel of R,u) by DefUnc; 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;
