reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;

theorem Th4:
  seq is decreasing iff for n,m st n<m holds seq.m<seq.n
proof
  thus seq is decreasing implies for n,m st n<m holds seq.m<seq.n
  proof
    assume seq is decreasing;
    then for n,k holds seq.(n+1+k)<seq.n by Th3;
    hence thesis by Lm3;
  end;
  assume
A1: for n,m st n<m holds seq.m<seq.n;
  let n,m;
  assume that
  n in dom seq and
  m in dom seq and
A2: n < m;
  thus thesis by A1,A2;
end;
