
theorem
  for A be 2-element set, a, b be Element of A st a <> b holds
    PrefSpace (A,a,b) is preference-like
  proof
    let A be 2-element set, a, b be Element of A;
    assume
Z1: a <> b;
    set X = PrefSpace (A,a,b);
a2: the PrefRel of X = {[a, b]} by Def3;
a3: the ToleranceRel of X = {[a, a], [b, b]} by Def3
    .= id A by Lemma4,Z1
    .= id (the carrier of X) by Def3;
    the PrefRel of X = {[a, b]} & the ToleranceRel of X = {[a, a], [b, b]} &
      the InternalRel of X = {}(A,A) by Def3; then
    (the PrefRel of X) /\ (the InternalRel of X) = {} &
      (the ToleranceRel of X) /\ (the InternalRel of X) = {} &
      (the PrefRel of X) /\ (the ToleranceRel of X) = {}
        by XBOOLE_0:def 7,Z1,Lemma8; then
A5: the PrefRel of X, the ToleranceRel of X, the InternalRel of X
      are_mutually_disjoint by XBOOLE_0:def 7;
C4: the PrefRel of X = {[a, b]} by Def3; then
C5: (the PrefRel of X)~ = {[b, a]} by Lemma7;
C6: the ToleranceRel of X = {[a, a], [b, b]} by Def3;
C1: the carrier of X = A by Def3;
D1: A = {a,b} by Z1,Lemma3;
    (the PrefRel of X) \/ (the PrefRel of X)~ \/
      (the ToleranceRel of X) \/
      (the InternalRel of X) = {[a, b]} \/ {[b, a]} \/ {[a, a], [b, b]} \/
        {}(A,A) by Def3, C4, C5, C6
      .= {[a, a], [a, b], [b, a], [b, b]} by Lemma6
      .= nabla the carrier of X by C1,D1,ZFMISC_1:122;
    hence thesis by a2, a3, A5,Def3,Lemma5,Z1;
  end;
