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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:75"/>
( <font color="Maroon">F</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font> implies <font color="Maroon">F</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">X</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">F</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:75_1"/>
for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st <font color="Olive">A1</font> is <a href="prob_1.html#V3">non-decreasing</a> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) holds <br/> <a href="kurato_2.html#K1">Union</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</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">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:75_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A1</font> is <a href="prob_1.html#V3">non-decreasing</a> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) )
 ;<br/>

<a NAME="E2:75_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T9" target="_self">Th9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A1</font> as    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">F</font> <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<a NAME="E4:75_1_1"/>
 <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46">PROB_1:46</a></i>;<br/>
<b>hence </b><a NAME="E5:75_1_1"/>
 <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:75_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">F</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#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E5:75_1"/>
<font color="Maroon">F</font> is <a href="prob_3.html#V3" target="_self">non-increasing-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="prob_3.html#D13" target="_self">PROB_3:def 13</a><br/>

<b>assume </b><a NAME="E1:75_1_2"/>
<font color="Maroon">A1</font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/>

<b>assume </b><a NAME="E2:75_1_2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:75_1_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E4:75_1_2"/>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:75_1"/>
<font color="Maroon">F</font> is    <a href="prob_3.html#M2" target="_self">MonotoneClass</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T9" target="_self">Th9</a>, <a class="ref" href="prob_3.html#T8" target="_self">Th8</a></i>;<br/>


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

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

<a NAME="E3:75"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> is <a href="prob_3.html#V3" target="_self">non-increasing-closed</a> &amp; <font color="Maroon">F</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#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:75"/>
( ( for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) &amp; ( for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</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">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</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">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:75_2_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">A2</font> =  <a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</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">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:75_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:75_2_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</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">k</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">E75</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:75_2_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:75_2_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>;<br/>
<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">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</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">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:75_2_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, <a class="ref" href="prob_3.html#T10" target="_self">Th10</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">E77</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<a NAME="E6:75_2_1"/>
 <a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font> is <a href="prob_1.html#V2">non-increasing</a>
 
<b>by </b><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">A1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T9" target="_self">Th9</a>, <a class="ref" href="prob_3.html#D2" target="_self">Def2</a>, <a class="ref" href="prob_3.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E8:75_2_1"/>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:75_2"/>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</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">F</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/>


</div>
