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 Th1:
  seq is increasing iff for n,m st n<m holds seq.n<seq.m
proof
  thus seq is increasing implies for n,m st n<m holds seq.n<seq.m
  proof
    assume seq is increasing;
    then for n holds seq.n<seq.(n+1);
    then for n,k holds seq.n<seq.(n+1+k) by Lm2;
    hence thesis by Lm2;
  end;
  assume
A1: for n,m st n<m holds seq.n<seq.m;
  let n,m;
  assume that
  n in dom seq and
  m in dom seq and
A2: n < m;
  thus thesis by A1,A2;
end;
