 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 :: 4.1 h)
  the InternalRel of R is reflexive total implies
    (f_1 R).the carrier of R = the carrier of R
  proof
    assume RR: the InternalRel of R is reflexive total;
A0: (f_1 R).([#]R) =
      { u where u is Element of R : (UncertaintyMap R).u meets [#]R } by Defff;
    the carrier of R c= { u where u is Element of R :
      (UncertaintyMap R).u meets [#]R }
    proof
      let y be object;
      assume y in the carrier of R; then
      reconsider u = y as Element of R;
      [u,u] in the InternalRel of R by LATTAD_1:1,RR; then
      u in (UncertaintyMap R).u by For3; then
      consider u being Element of R such that
A1:   u = y & (UncertaintyMap R).u meets [#]R by XBOOLE_0:3;
      thus thesis by A1;
    end;
    hence thesis by A0;
  end;
