<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>thus </b><a NAME="E1:77"/>
( <font color="Maroon">seq</font> is <a href="seqm_3.html#V1" target="_self">increasing</a> &amp; <font color="Maroon">seq1</font> is <a href="seqm_3.html#V1" target="_self">increasing</a> implies <font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font> is <a href="seqm_3.html#V1" target="_self">increasing</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:77_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V1" target="_self">increasing</a>
 <b>and </b><br/><a NAME="E2:77_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V1" target="_self">increasing</a>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D11" target="_self">SEQM_3:def 11</a><br/>

<a NAME="E3:77_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E4:77_1"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E10"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<a NAME="E5:77_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E11"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:77_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:77_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:77"/>
( <font color="Maroon">seq</font> is <a href="seqm_3.html#V2" target="_self">decreasing</a> &amp; <font color="Maroon">seq1</font> is <a href="seqm_3.html#V2" target="_self">decreasing</a> implies <font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font> is <a href="seqm_3.html#V2" target="_self">decreasing</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:77_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V2" target="_self">decreasing</a>
 <b>and </b><br/><a NAME="E2:77_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V2" target="_self">decreasing</a>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D12" target="_self">SEQM_3:def 12</a><br/>

<a NAME="E3:77_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E4:77_2"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E10"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<a NAME="E5:77_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E11"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:77_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:77_2"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:77"/>
( <font color="Maroon">seq</font> is <a href="seqm_3.html#V3" target="_self">non-decreasing</a> &amp; <font color="Maroon">seq1</font> is <a href="seqm_3.html#V3" target="_self">non-decreasing</a> implies <font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font> is <a href="seqm_3.html#V3" target="_self">non-decreasing</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:77_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V3" target="_self">non-decreasing</a>
 <b>and </b><br/><a NAME="E2:77_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V3" target="_self">non-decreasing</a>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D13" target="_self">SEQM_3:def 13</a><br/>

<a NAME="E3:77_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E4:77_3"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E10"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<a NAME="E5:77_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E11"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:77_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:77_3"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E4:77"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V4" target="_self">non-increasing</a>
 <b>and </b><br/><a NAME="E5:77"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V4" target="_self">non-increasing</a>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D14" target="_self">SEQM_3:def 14</a><br/>

<a NAME="E6:77"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E7:77"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E10"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<a NAME="E8:77"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="seqm_3.html#E11"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:77"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E10:77"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div>
