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<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="firebrick">::  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#M2" title="VALUED_0:mode.2">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="E1:41"/><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="E2:41"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c2">seq1</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c1">seq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="41_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_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="firebrick">::  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="E2:41_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="#E2:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="seqm_3.html#T25" target="_self" title="SEQM_3:th.25">Th52</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_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#D5" target="_self" title="SEQM_3:def.5">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="41_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_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="firebrick">::  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="E2:41_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="#E2:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="seqm_3.html#T24" target="_self" title="SEQM_3:th.24">Th51</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_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#D5" target="_self" title="SEQM_3:def.5">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:41"/>
<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="#E1:41"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="seqm_3.html#D5" target="_self" title="SEQM_3:def.5">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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