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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a>  &amp; <font color="Maroon" title="c2">seq1</font> is    <a href="valued_0.html#M1" title="VALUED_0: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#V1" title="SEQM_3:attr.1">monotone</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> 
 <span class="kw">and </span><br/><a NAME="E2:69"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c2">seq1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> </span><br/>

<div><span class="lab"><font color="Green" title="E35">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:69_1"/>
<font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> </span><br/><a NAME="E2:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq1</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="seqm_3.html#T52" target="_self" title="SEQM_3:th.52">Th52</a></span>;<br/><span class="kw">hence </span><a NAME="E3:69_1"/>
<font color="Maroon" title="c2">seq1</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#D7" target="_self" title="SEQM_3:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:69_2"/>
<font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> </span><br/><a NAME="E2:69_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">seq1</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="seqm_3.html#T51" target="_self" title="SEQM_3:th.51">Th51</a></span>;<br/><span class="kw">hence </span><a NAME="E3:69_2"/>
<font color="Maroon" title="c2">seq1</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#D7" target="_self" title="SEQM_3:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:69"/>
<font color="Maroon" title="c2">seq1</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="seqm_3.html#D7" target="_self" title="SEQM_3:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
