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

theorem Th94:
  for A being Preorder, B being Subset of A,
    s1 being FinSequence of A
  st
    s1 is B-asc_ordering
  holds
    ex s2 being FinSequence of QuotientOrder(A) st
      s2 is ((proj A).:B)-asc_ordering
proof
  let A be Preorder, B be Subset of A;
  let s1 be FinSequence of the carrier of A;
  assume A1: s1 is B-asc_ordering;
  then A2: the InternalRel of A is_connected_in B by Th83;
  reconsider B as finite Subset of A by A1;
  (proj A).:B is finite;
  hence thesis by A2, Th93, Th89;
end;
