
theorem
  for P being non empty PreferenceStr st P is preference-like holds
    the ToleranceRel of P = CharRel P /\ (CharRel P)~
  proof
    let P be non empty PreferenceStr;
    assume
A1: P is 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 T iff [x,y] in C /\ C~
    proof
      let x, y be object;
Z1:   [x,y] in T implies [x,y] in C /\ C~
      proof
        assume
A3:     [x,y] in T; then
    A2: x in field T & y in field T by RELAT_1:15;
        [x,y] in T & [y,x] in T by A1,A3, A2, RELAT_2:def 3,def 11; then
        [x,y] in R \/ T & [y,x] in R \/ T by XBOOLE_0:def 3; then
        [x,y] in C & [x,y] in C~ by RELAT_1:def 7;
        hence thesis by XBOOLE_0:def 4;
      end;
      [x,y] in C /\ C~ implies [x,y] in T
      proof
        assume [x,y] in C /\ C~; then
        [x,y] in C & [x,y] in C~ by XBOOLE_0:def 4; then
        [x,y] in R \/ T & [y,x] in R \/ T by RELAT_1:def 7; then
        ([x,y] in R & [y,x] in R) or ([x,y] in R & [y,x] in T) or
          ([x,y] in T & [y,x] in R) or ([x,y] in T & [y,x] in T)
          by XBOOLE_0:def 3;
        hence [x,y] in T by LemSym,A1,LemAsym;
      end;
      hence thesis by Z1;
    end;
    hence thesis by RELAT_1:def 2;
  end;
