
theorem
  for P being non empty PIStr st P is PI-preference-like holds
    the PrefRel of P = CharRel P /\ ((CharRel P)~)`
  proof
    let P be non empty PIStr;
    assume
A1: P is PI-preference-like;
    set R = the PrefRel of P, T = the ToleranceRel of P, C = CharRel P;
    for x, y being object holds [x,y] in R iff [x,y] in C /\ (C~)`
    proof
      let x, y be object;
B1:   [x,y] in R implies [x,y] in C /\ (C~)`
      proof
        assume
Z0:     [x,y] in R; then
k1:     x in field R & y in field R by RELAT_1:15;
Z1:     not [x,y] in T by Z0, XBOOLE_0:def 4,A1;
        ([x,y] in R or [x,y] in T) & not [y,x] in R & not [y,x] in T
          by LemAsym,Z0,A1,Z1,LemSym; then
cc:     [x,y] in (R \/ T) & not [y,x] in (R \/ T) by XBOOLE_0:def 3; then
        [x,y] in C & not [x,y] in C~ by RELAT_1:def 7; then
        [x,y] in (C~)` by Lemma12b,k1;
        hence thesis by XBOOLE_0:def 4,cc;
      end;
      [x,y] in C /\ (C~)` implies [x,y] in R
      proof
        assume
cc:     [x,y] in C /\ (C~)`; then
        [x,y] in C & [x,y] in (C~)` by XBOOLE_0:def 4; then
        not [x,y] in C~ by XBOOLE_0:def 5; then
        [x,y] in C & not [y,x] in C by RELAT_1:def 7,cc,XBOOLE_0:def 4; then
        [x,y] in R & not [y,x] in R & not [y,x] in T or [x,y] in T &
          not [y,x] in R & not [y,x] in T by XBOOLE_0:def 3;
        hence thesis by LemSym,A1;
      end;
      hence thesis by B1;
    end;
    hence thesis by RELAT_1:def 2;
  end;
