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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <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/>



<b>thus </b><a NAME="E1:75"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<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="75_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:75_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>1</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="kurato_2.html#K1">Union</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="75_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>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:75_1_1"/><i><font color="Green">E64</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/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="prob_3.html#E1:75_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<a NAME="E3:75_1_1"/>
 <a href="kurato_2.html#K1">Union</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="ref" href="prob_1.html#T46">PROB_1:46</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1_1"/>
 <a href="kurato_2.html#K1">Union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:75_1"/><b>then </b><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>
 
<b>by </b><i><a class="ref" href="prob_3.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E5:75_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>1</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="prob_1.html#K4">Intersection</a> <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="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/>
<a NAME="E6:75_1"/><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#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E7:75_1"/>
<font color="Maroon">c<sub>2</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#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="prob_3.html#D14" target="_self">Def14</a></i>;<br/>


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

<b>assume </b><a NAME="E2:75"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:75"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="prob_3.html#V3" target="_self">non-increasing-closed</a> &amp; <font color="Maroon">c<sub>2</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#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:75"/>
( ( 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 ( 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>2</sub></font> ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( 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>2</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>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:75_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 ( 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>2</sub></font> ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:75_2_1"/><i><font color="Green">E65</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>3</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>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:75_2_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> 0
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:75_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:75_2_1"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E4:75_2_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:75_2_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:75_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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#E1:75_2_1"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E3:75_2_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:75_2_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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#E1:75_2_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="prob_3.html#E2:75_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:75_2_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="prob_3.html#E3:75_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="prob_3.html#E4:75_2_1"><i><font color="Green">E67</font></i></a>);<br/></i>
<a NAME="E6:75_2_1"/>
 <a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E7:75_2_1"/><b>then </b>
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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#E3:75"><i><font color="Green">E64</font></i></a>, <a class="txt" href="prob_3.html#E5:75_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="prob_3.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E8:75_2_1"/>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>3</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_3.html#T16" target="_self">Th16</a></i>;<br/>


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

<b>thus </b><a NAME="E2:75_2"/>
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>2</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>2</sub></font>
 <b>by </b><i><a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:75"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/>


</div>
