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<b>let </b><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c2">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> implies <font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a>
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a></span><br/>

<div><i><font color="Green" title="E45">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a></span><br/><a NAME="E2:66_1"/><b>then </b>
<font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:66"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="seqm_3.html#T51" target="_self" title="SEQM_3:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E3:66_1"/>
<font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D7" target="_self" title="SEQM_3:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2"/>
<font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a></span><br/><a NAME="E2:66_2"/><b>then </b>
<font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E2:66"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="seqm_3.html#T52" target="_self" title="SEQM_3:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:66_2"/>
<font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D7" target="_self" title="SEQM_3:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:66"/>
<font color="Maroon" title="c2">seq1</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1:66"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="seqm_3.html#E3:66"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="seqm_3.html#D7" target="_self" title="SEQM_3:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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