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<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>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V5" target="_self">monotone</a>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is    <a href="seqm_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">seq</font>
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1"/>
<font color="Maroon">seq</font> is <a href="seqm_3.html#V3" target="_self">non-decreasing</a>
 ;<br/><a NAME="E2:67_1"/><b>then </b>
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V3" target="_self">non-decreasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D2" target="_self">Def2</a>, </i>;<br/><b>hence </b><a NAME="E3:67_1"/>
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V5" target="_self">monotone</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_2"/>
<font color="Maroon">seq</font> is <a href="seqm_3.html#V4" target="_self">non-increasing</a>
 ;<br/><a NAME="E2:67_2"/><b>then </b>
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V4" target="_self">non-increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D2" target="_self">Def2</a>, <a class="txt" href="seqm_3.html#E5"><i><font color="Green">Lemma43</font></i></a></i>;<br/><b>hence </b><a NAME="E3:67_2"/>
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V5" target="_self">monotone</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:67"/>
<font color="Maroon">seq1</font> is <a href="seqm_3.html#V5" target="_self">monotone</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D1" target="_self">Def1</a>, <a class="ref" href="seqm_3.html#D3" target="_self">Def3</a>, </i>;<br/>


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