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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:79"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_3.html#D15" target="_self">Def15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xboole_0.html#K3">/\</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="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E2:79_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</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="79_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:79_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:79_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K8">/\</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#E1:79_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:79_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</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#E3:79_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:79"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  holds <br/><font color="Olive">b<sub>2</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:79_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:79_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</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#E3:79"><i><font color="Green">E69</font></i></a>, <a class="txt" href="prob_3.html#E1:79_2"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E2:79_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:79_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:79_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</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>1</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="kurato_2.html#K7">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</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>5</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</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="kurato_2.html#K7">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</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>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:79_2_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="setlim_1.html#V1">monotone</a>
 <b>and </b><br/><a NAME="E2:79_2_2_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E3:79_2_2_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</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="79_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:79_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:79_2_2_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:79_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</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#E1:79_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="prob_3.html#E3:79"><i><font color="Green">E69</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:79_2_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="kurato_2.html#V3">convergent</a>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_2_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="setlim_1.html#T71">SETLIM_1:71</a></i>;<br/>
<a NAME="E5:79_2_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="setlim_2.html#K8">(\)</a> <font color="Maroon">c<sub>4</sub></font> is <a href="setlim_1.html#V1">monotone</a>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_2_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="setlim_2.html#T32">SETLIM_2:32</a></i>;<br/>
<a NAME="E6:79_2_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="setlim_2.html#K8">(\)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="kurato_2.html#K7">.</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="79_2_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:79_2_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:79_2_2_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E2:79_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K7">\</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#E1:79_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="prob_3.html#E3:79"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:79_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="setlim_2.html#K8">(\)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="setlim_2.html#D8">SETLIM_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:79_2_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="setlim_1.html#K3">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="setlim_2.html#K8">(\)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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#E5:79_2_2_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E8:79_2_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K7">(\)</a> <font color="Maroon">c<sub>6</sub></font> is <a href="setlim_1.html#V1">monotone</a>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_2_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="setlim_2.html#T29">SETLIM_2:29</a></i>;<br/>
<a NAME="E9:79_2_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K7">(\)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="kurato_2.html#K7">.</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="79_2_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:79_2_2_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:79_2_2_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E2:79_2_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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#E1:79_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="prob_3.html#E3:79"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:79_2_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K7">(\)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="setlim_2.html#D7">SETLIM_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:79_2_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="setlim_1.html#K3">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K7">(\)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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#E8:79_2_2_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E11:79_2_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K5">(/\)</a> <font color="Maroon">c<sub>6</sub></font> is <a href="setlim_1.html#V1">monotone</a>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_2_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="setlim_2.html#T23">SETLIM_2:23</a></i>;<br/>
<a NAME="E12:79_2_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K5">(/\)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="kurato_2.html#K7">.</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="79_2_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:79_2_2_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:79_2_2_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E2:79_2_2_1_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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#E1:79_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="prob_3.html#E3:79"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:79_2_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K5">(/\)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="setlim_2.html#D5">SETLIM_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:79_2_2_1"/><b>then </b>
 <a href="setlim_1.html#K3">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setlim_2.html#K5">(/\)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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#E11:79_2_2_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E14:79_2_2_1"/><b>then </b>
(  <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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#E3:79_2_2_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="prob_3.html#E7:79_2_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="prob_3.html#E4:79_2_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="setlim_2.html#T95">SETLIM_2:95</a>, <a class="txt" href="prob_3.html#E10:79_2_2_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="setlim_2.html#T94">SETLIM_2:94</a>, <a class="ref" href="setlim_2.html#T92">SETLIM_2:92</a>, <a class="txt" href="prob_3.html#E1:79_2_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="prob_3.html#T74" target="_self">Th74</a></i>;<br/>
<b>hence </b><a NAME="E15:79_2_2_1"/>
 <a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:79_2"><i><font color="Green">E71</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</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>1</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="kurato_2.html#K7">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</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>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="prob_3.html#E2:79_2"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:79_2"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="prob_3.html#T74" target="_self">Th74</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:79"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:79_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:79_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:79_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:79_3_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T12">PROB_1:12</a></i>;<br/>
<a NAME="E3:79_3_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="prob_1.html#T12">PROB_1:12</a></i>;<br/>
<a NAME="E4:79_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:79_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:79_3_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="prob_3.html#E2:79_3_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="prob_3.html#E3:79_3_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="prob_3.html#E1:79_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="prob_3.html#E1:79"><i><font color="Green">E68</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:79_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E4:79_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79"><i><font color="Green">E68</font></i></a>, <a class="txt" href="prob_3.html#E1:79_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="prob_3.html#E4:79"><i><font color="Green">E70</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:79_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:79_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="prob_3.html#D15" target="_self">Def15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:79"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:79_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:79_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:79_4_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<a NAME="E2:79_4_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_4_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="prob_3.html#E5:79"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E3:79_4_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E4:79_4_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</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#E2:79_4_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="prob_3.html#E3:79"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:79_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:79_4_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="prob_3.html#E1:79"><i><font color="Green">E68</font></i></a>, <a class="txt" href="prob_3.html#E1:79_4"><i><font color="Green">E73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:79_4"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="prob_3.html#E4:79"><i><font color="Green">E70</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:79_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:79_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="prob_3.html#D15" target="_self">Def15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:79"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="finsub_1.html#V2">cap-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:79_5_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/><a NAME="E2:79_5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="prob_3.html#E1:79_5_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="prob_3.html#E6:79"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E3:79_5_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</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#E3:79"><i><font color="Green">E69</font></i></a>, <a class="txt" href="prob_3.html#E1:79_5_1"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_5"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="finsub_1.html#V2">cap-closed</a>
 <b>by </b><i><a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:79"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  <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="79_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:79_6"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<a NAME="E2:79_6"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T11">PROB_1:11</a></i>;<br/>
<a NAME="E3:79_6"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>1</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#E1:79"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E4:79_6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : ( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:79_6"><i><font color="Green">E74</font></i></a>, <a class="txt" href="prob_3.html#E6:79"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E5:79_6"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</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#E3:79"><i><font color="Green">E69</font></i></a>, <a class="txt" href="prob_3.html#E3:79_6"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:79_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:79"/>
 <a href="prob_3.html#K14" target="_self">monotoneclass</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/>


</div>
