reserve X,Z for set;
reserve x,y,z for object;
reserve A,B,C for Subset of X;

theorem
  for A being Preorder, B being Subset of A,
    s1 being FinSequence of A
  st
    s1 is B-desc_ordering
  holds
    ex s2 being FinSequence of QuotientOrder(A) st
      s2 is ((proj A).:B)-desc_ordering
proof
  let A be Preorder, B be Subset of A;
  let s1 be FinSequence of the carrier of A;
  assume s1 is B-desc_ordering;
  then Rev(Rev(s1)) is B-desc_ordering;
  then Rev(s1) is B-asc_ordering by Th75;
  then consider s2 being FinSequence of QuotientOrder(A) such that
    A1: s2 is ((proj A).:B)-asc_ordering by Th94;
  take Rev(s2);
  thus thesis by A1, Th75;
end;
