reserve n,m for Nat;

theorem Th23:
  for f being real-valued FinSequence st
  f is non-decreasing holds -f is non-increasing
proof
  let f be real-valued FinSequence;
  assume
A1: f is non-decreasing;
  for n being Nat st n in dom (-f) & n+1 in dom (-f) holds (-f)
  .n >= (-f).(n+1)
  proof
    let n be Nat;
A2: dom (-f)=dom f by VALUED_1:8;
A3: (-f).n=-(f.n) & (-f).(n+1)= -(f.(n+1)) by RVSUM_1:17;
    assume n in dom (-f) & n+1 in dom (-f);
    then f.n <= f.(n+1) by A1,A2;
    hence thesis by A3,XREAL_1:24;
  end;
  hence thesis by RFINSEQ:def 3;
end;
