reserve x,y for set,
  n,m for Nat,
  r,s for Real;

theorem Th19:
  for R be real-valued FinSequence holds R is non-increasing iff
   for n,m be Nat st n in dom R & m in dom R & n<m holds R.n>=R.m
proof
  let R be real-valued FinSequence;
  thus R is non-increasing implies
   for n,m being Nat st n in dom R & m in dom R & n<m
  holds R.n>=R.m;
  assume
A1: for n,m being Nat st n in dom R & m in dom R & n<m holds R.n>=R.m;
  let n;
  n<n+1 by NAT_1:13;
  hence thesis by A1;
end;
