 reserve f,g for Function;
 reserve R for non empty reflexive RelStr;
 reserve R for non empty RelStr;
 reserve f for Function of the carrier of R, bool the carrier of R;

theorem
  R is reflexive implies
    union ((UncertaintyMap R).:[#]R) = the carrier of R
  proof
    assume AA: R is reflexive;
B1: the carrier of R c= union ((UncertaintyMap R).:[#]R)
    proof
      let t be object;
      assume t in the carrier of R; then
      reconsider tt = t as Element of R;
      dom UncertaintyMap R = the carrier of R by FUNCT_2:def 1; then
A2:   tt in dom UncertaintyMap R & tt in [#]R;
A3:   tt in (UncertaintyMap R).t by AA,ReflUnc;
      (UncertaintyMap R).t in (UncertaintyMap R).:[#]R by A2,FUNCT_1:def 6;
      hence thesis by TARSKI:def 4,A3;
    end;
    union ((UncertaintyMap R).:[#]R) c= the carrier of R
    proof
      let f be object;
      assume f in union ((UncertaintyMap R).:[#]R); then
      ex tt being set st
      f in tt & tt in ((UncertaintyMap R).:[#]R) by TARSKI:def 4;
      hence thesis;
    end;
    hence thesis by B1;
  end;
