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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c3">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ;<br/>

<a NAME="E2:27"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K6" title="SETLIM_2:func.6">(\/)</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K6" title="SETLIM_2:func.6">(\/)</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 ;<br/>

<a NAME="E2:27_1"/>
<font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="setlim_2.html#E1:27_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="setlim_2.html#E1:27"><i><font color="Green" title="E33">A1</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="E3:27_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E4:27_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K6" title="SETLIM_2:func.6">(\/)</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="setlim_2.html#D6" target="_self" title="SETLIM_2:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E5:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K6" title="SETLIM_2:func.6">(\/)</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K6" title="SETLIM_2:func.6">(\/)</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="setlim_2.html#D6" target="_self" title="SETLIM_2:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27"/>
<font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K6" title="SETLIM_2:func.6">(\/)</a> <font color="Maroon" title="c3">A1</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/>


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