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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:74"/>
( <font color="Maroon">c<sub>1</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>2</sub></font> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) holds <br/> <a href="setlim_1.html#K3">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:74_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="prob_3.html#V2" target="_self">non-decreasing-closed</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="prob_3.html#V3" target="_self">non-increasing-closed</a> )
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:74_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="prob_3.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:74_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) holds <br/> <a href="setlim_1.html#K3">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:74_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="prob_1.html#V3">non-decreasing</a> or <font color="Maroon">c<sub>4</sub></font> is <a href="prob_1.html#V2">non-increasing</a> )
 </span><b>by </b><i><a class="txt" href="prob_3.html#E1:74_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="setlim_1.html#D1">SETLIM_1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="prob_1.html#V3">non-decreasing</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1_1"/>
 <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E3:74_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="prob_3.html#E1:74_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="prob_3.html#T71" target="_self">Th71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1_2"/>
 <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E3:74_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="prob_3.html#E1:74_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="prob_3.html#T72" target="_self">Th72</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:74_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) holds <br/> <a href="setlim_1.html#K3">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) )
 ;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:74"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:74"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) holds <br/> <a href="setlim_1.html#K3">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="prob_3.html#E2:74"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E5:74"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="prob_1.html#V3">non-decreasing</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) holds <br/> <a href="setlim_1.html#K3">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:74_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="prob_1.html#V3">non-decreasing</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<a NAME="E2:74_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) implies  <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="prob_3.html#E3:74"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:74_2"/>
 <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:74_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="setlim_1.html#D1">SETLIM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:74"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="prob_3.html#V2" target="_self">non-decreasing-closed</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T71" target="_self">Th71</a></i>;<br/>
<a NAME="E7:74"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="prob_1.html#V2">non-increasing</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) holds <br/> <a href="setlim_1.html#K3">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:74_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="prob_1.html#V2">non-increasing</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<a NAME="E2:74_3"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="setlim_1.html#V1">monotone</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) implies  <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="prob_3.html#E3:74"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:74_3"/>
 <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:74_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="setlim_1.html#D1">SETLIM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:74"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="prob_3.html#V3" target="_self">non-increasing-closed</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T72" target="_self">Th72</a></i>;<br/>
<b>hence </b><a NAME="E9:74"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E6:74"><i><font color="Green">E65</font></i></a>, <a class="ref" href="prob_3.html#D14" target="_self">Def14</a></i>;<br/>


</div>
