 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 b)
  R is reflexive implies
    Flip f_1 R cc= id bool the carrier of R
  proof
    assume TT: R is reflexive;
    set f = Flip f_1 R;
    set g = id bool the carrier of R;
A1: dom f c= dom g;
    for i being set st i in dom f holds f.i c= g.i
    proof
      let i be set;
      assume i in dom f; then
      reconsider ii = i as Subset of R;
      { w where w is Element of R : (UncertaintyMap R).w c= ii } c= ii
      proof
        let r be object;
        assume r in { w where w is Element of R : (UncertaintyMap R).w c= ii };
        then consider w being Element of R such that
C4:     r = w & (UncertaintyMap R).w c= ii;
        thus thesis by C4,TT,ReflUnc;
      end;
      hence thesis by FlipF1;
    end;
    hence thesis by A1,ALTCAT_2:def 1;
  end;
