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

theorem :: 4.1 b)
  the InternalRel of R is total reflexive implies
    id bool the carrier of R cc= f_1 R
  proof
    assume zz: the InternalRel of R is total reflexive;
    set f = id bool the carrier of R;
    set g = f_1 R;
A1: dom f c= dom g by FUNCT_2:def 1;
    for i being set st i in dom f holds f.i c= g.i
    proof
      let i be set;
      assume k2: i in dom f; then
      reconsider ii = i as Subset of R;
      i c= { u where u is Element of R : (UncertaintyMap R).u meets ii }
      proof
        let y be object;
        assume D1: y in i; then
        reconsider wy = y as Element of R by k2;
        [wy,wy] in the InternalRel of R by zz,LATTAD_1:1; then
        wy in (UncertaintyMap R).wy by For3; then
        (UncertaintyMap R).wy meets ii by XBOOLE_0:3,D1;
        hence thesis;
      end;
      hence f.i c= g.i by Defff;
    end;
    hence thesis by A1,ALTCAT_2:def 1;
  end;
