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

theorem Th75:
  for A being RelStr, B being Subset of A, s being FinSequence of A holds
    s is B-asc_ordering iff Rev(s) is B-desc_ordering
proof
  let A be RelStr;
  let B be Subset of A;
  let s be FinSequence of A;
  hereby
    assume A1: s is B-asc_ordering;
    then A2: Rev(s) is weakly-descending by Th73;
    rng Rev(s) = B by A1, FINSEQ_5:57;
    hence Rev(s) is B-desc_ordering by A2, A1;
  end;
  assume A4: Rev(s) is B-desc_ordering;
  then A5: s is weakly-ascending by Th73;
  A6: rng s = B by A4, FINSEQ_5:57;
  Rev(Rev(s)) is one-to-one by A4;
  hence s is B-asc_ordering by A5, A6;
end;
