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;
reserve Nseq for increasing sequence of NAT;

theorem Th34:
  (seq is increasing & seq1 is increasing implies seq+seq1 is
  increasing) & (seq is decreasing & seq1 is decreasing implies seq+seq1 is
decreasing) & (seq is non-decreasing & seq1 is non-decreasing implies seq+seq1
  is non-decreasing) & (seq is non-increasing & seq1 is non-increasing implies
  seq+seq1 is non-increasing)
proof
  thus seq is increasing & seq1 is increasing implies seq+seq1 is increasing
  proof
    assume
A1: seq is increasing & seq1 is increasing;
    let n;
    seq.n<seq.(n+1) & seq1.n<seq1.(n+1) by A1;
    then seq.n+seq1.n<seq.(n+1)+seq1.(n+1) by XREAL_1:8;
    then (seq+seq1).n<seq.(n+1)+seq1.(n+1) by SEQ_1:7;
    hence thesis by SEQ_1:7;
  end;
  thus seq is decreasing & seq1 is decreasing implies seq+seq1 is decreasing
  proof
    assume
A2: seq is decreasing & seq1 is decreasing;
    let n;
    seq.(n+1)<seq.n & seq1.(n+1)<seq1.n by A2;
    then seq.(n+1)+seq1.(n+1)<seq.n+seq1.n by XREAL_1:8;
    then (seq+seq1).(n+1)<seq.n+seq1.n by SEQ_1:7;
    hence thesis by SEQ_1:7;
  end;
  thus seq is non-decreasing & seq1 is non-decreasing implies seq+seq1 is
  non-decreasing
  proof
    assume
A3: seq is non-decreasing & seq1 is non-decreasing;
    let n;
    seq.n<=seq.(n+1) & seq1.n<=seq1.(n+1) by A3;
    then seq.n+seq1.n<=seq.(n+1)+seq1.(n+1) by XREAL_1:7;
    then (seq+seq1).n<=seq.(n+1)+seq1.(n+1) by SEQ_1:7;
    hence thesis by SEQ_1:7;
  end;
  assume
A4: seq is non-increasing & seq1 is non-increasing;
  let n;
  seq.(n+1)<=seq.n & seq1.(n+1)<=seq1.n by A4;
  then seq.(n+1)+seq1.(n+1)<=seq.n+seq1.n by XREAL_1:7;
  then (seq+seq1).(n+1)<=seq.n+seq1.n by SEQ_1:7;
  hence thesis by SEQ_1:7;
end;
