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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font> iff <font color="Olive" title="b1">F</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font> iff <font color="Maroon" title="c2">F</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font> )</span><br/>



<b>thus </b><a NAME="E1:76"/>
( <font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">F</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1"/>
<font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c2">F</font> as   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:76_1"/>
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> holds <br/> <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> implies  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<b>assume </b><a NAME="E1:76_1_1"/><i><font color="Green" title="E63">A1</font></i>: 
( <font color="Maroon" title="c4">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">A2</font> = <font color="Maroon" title="c4">A1</font> as    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c3">F</font> <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a>, <a class="txt" href="#E1:76_1_1"><i><font color="Green" title="E63">A1</font></i></a></i>;<br/>
<a NAME="E3:76_1_1"/>
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c5">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46" title="PROB_1:th.46">PROB_1:46</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_1"/>
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:76_1"/><b>then </b><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="prob_3.html#V2" title="PROB_3:attr.2">non-decreasing-closed</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#D12" target="_self" title="PROB_3:def.12">Def12</a></i>;<br/>
<a NAME="E5:76_1"/>
<font color="Maroon" title="c3">F</font> is <a href="prob_3.html#V3" title="PROB_3:attr.3">non-increasing-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="prob_3.html#D13" target="_self" title="PROB_3:def.13">PROB_3:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A1</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> implies  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<b>assume </b><a NAME="E1:76_1_2"/>
<font color="Maroon" title="c4">A1</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> or  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<b>assume </b><a NAME="E2:76_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font></span><br/>

<b>hence </b><a NAME="E3:76_1_2"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:76_1"/>
<font color="Maroon" title="c2">F</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E4:76_1"><i><font color="Green" title="E63">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:76"/>
<font color="Maroon" title="c2">F</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E3:76"/><b>then </b><i><font color="Green" title="E63">A3</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="prob_3.html#V3" title="PROB_3:attr.3">non-increasing-closed</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_3.html#V2" title="PROB_3:attr.2">non-decreasing-closed</a> )
 
;<br/>
<a NAME="E4:76"/>
( ( for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> ) &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:76_2"/>
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1"><b>proof </b></a><div class="add">

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> implies  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:76_2_1"/><i><font color="Green" title="E64">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<b>set </b><font color="Maroon" title="c4">A2</font> =  <a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>;<br/>
<a NAME="E2:76_2_1"/><i><font color="Green" title="E65">X</font></i>: 
<font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">A1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<a NAME="E3:76_2_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self" title="PROB_3:def.1">Def1</a></i>;<br/>
<a NAME="E4:76_2_1"/><b>then </b><i><font color="Green" title="E66">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="#E1:76_2_1"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="#E2:76_2_1"><i><font color="Green" title="E65">X</font></i></a></i>;<br/>
<a NAME="E5:76_2_1"/><i><font color="Green" title="E67">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:76_2_1_1"/><i><font color="Green" title="E68">A7</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:76_2_1_1"/>
<font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">A1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<a NAME="E3:76_2_1_1"/><b>then </b><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E1:76_2_1"><i><font color="Green" title="E64">A4</font></i></a></i>;<br/>
<a NAME="E4:76_2_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self" title="PROB_3:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:76_2_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E1:76_2_1_1"><i><font color="Green" title="E68">A7</font></i></a>, <a class="txt" href="#E3:76_2_1_1"><i><font color="Green" title="E69">A8</font></i></a>, <a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:76_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:76_2_1"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="#E5:76_2_1"><i><font color="Green" title="E67">A6</font></i></a>);<br/></i>
<a NAME="E7:76_2_1"/><b>then </b><i><font color="Green" title="E68">X</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/>
<a NAME="E8:76_2_1"/>
 <a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T12" target="_self" title="PROB_3:th.12">Th12</a></i>;<br/>
<a NAME="E9:76_2_1"/><b>then </b>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E3:76"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="prob_3.html#D13" target="_self" title="PROB_3:def.13">Def13</a>, <a class="txt" href="#E7:76_2_1"><i><font color="Green" title="E68">X</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:76_2_1"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T16" target="_self" title="PROB_3:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:76_2"/>
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:76"/>
<font color="Maroon" title="c2">F</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
