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
  (seq is increasing & seq1 is constant implies seq+seq1 is increasing)
& (seq is decreasing & seq1 is constant implies seq+seq1 is decreasing) & (seq
is non-decreasing & seq1 is constant implies seq+seq1 is non-decreasing) & (seq
  is non-increasing & seq1 is constant implies seq+seq1 is non-increasing)
proof
  thus seq is increasing & seq1 is constant implies seq+seq1 is increasing
  proof
    assume that
A1: seq is increasing and
A2: seq1 is constant;
    let n;
    consider r1 being Element of REAL such that
A3: for n being Nat holds seq1.n=r1 by A2;
    seq.n<seq.(n+1) by A1;
    then seq.n+r1<seq.(n+1)+r1 by XREAL_1:6;
    then seq.n+seq1.n<seq.(n+1)+r1 by A3;
    then seq.n+seq1.n<seq.(n+1)+seq1.(n+1) by A3;
    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 constant implies seq+seq1 is decreasing
  proof
    assume that
A4: seq is decreasing and
A5: seq1 is constant;
    let n;
    consider r1 being Element of REAL such that
A6: for n being Nat holds seq1.n=r1 by A5;
    seq.(n+1)<seq.n by A4;
    then seq.(n+1)+r1<seq.n+r1 by XREAL_1:6;
    then seq.(n+1)+seq1.(n+1)<seq.n+r1 by A6;
    then seq.(n+1)+seq1.(n+1)<seq.n+seq1.n by A6;
    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 constant implies seq+seq1 is
  non-decreasing by Th34;
  assume seq is non-increasing & seq1 is constant;
  hence thesis by Th34;
end;
