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

theorem
  for A being transitive RelStr, B, C being Subset of A,
    s1 being FinSequence of A, x being Element of A
  st s1 is C-desc_ordering & not x in C & B = C \/ {x} &
    for y being Element of A st y in C holds x <= y
  holds
    ex s2 being FinSequence of A st
      s2 = s1 ^ <*x*> & s2 is B-desc_ordering
proof
  let A be transitive RelStr;
  let B,C be Subset of A;
  let s1 be FinSequence of A;
  let x be Element of A;
  assume that
    A1: s1 is C-desc_ordering and
    A2: not x in C & B = C \/ {x} &
      for y being Element of A st y in C holds x <= y;
  Rev(Rev(s1)) is C-desc_ordering by A1;
  then Rev(s1) is C-asc_ordering by Th75;
  then consider s2 being FinSequence of A such that
    A3: s2 = <*x*> ^ Rev(s1) and
    A4: s2 is B-asc_ordering by A2, Th85;
  take Rev(s2);
  thus Rev(s2) = Rev(Rev(s1)) ^ Rev(<*x*>) by A3, FINSEQ_5:64
    .= s1 ^ <*x*> by FINSEQ_5:60;
  thus Rev(s2) is B-desc_ordering by A4, Th75;
end;
