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<b>let </b><font color="Maroon" title="c1">Omega</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">ASeq</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>;<br/>



<b>thus </b><a NAME="E1:13"/>
( <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a> implies  <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ;<br/><a NAME="E2:13_1_1"/><b>then </b>
<font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="prob_2.html#E1:13_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prob_1.html#D6" title="PROB_1:def.6">PROB_1:def 6</a></i>;<br/><a NAME="E3:13_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></i>;<br/><a NAME="E4:13_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:13_1_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1"/>
 <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="prob_1.html#D7" title="PROB_1:def.7">PROB_1:def 7</a></i>;<br/>


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

<b>assume </b><a NAME="E2:13"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:13_2"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ;<br/><a NAME="E2:13_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="prob_2.html#E2:13"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="prob_1.html#D7" title="PROB_1:def.7">PROB_1:def 7</a></i>;<br/><a NAME="E3:13_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</a></i>;<br/><a NAME="E4:13_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:13_2"/>
<font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13"/>
<font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 <b>by </b><i><a class="ref" href="prob_1.html#D6" title="PROB_1:def.6">PROB_1:def 6</a></i>;<br/>


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