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

<b>let </b><font color="Maroon">Omega</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">X</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font>;<br/>



<b>set </b><font color="Maroon">V</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> ;<br/>
<a NAME="E1:67"/><i><font color="Green">E33</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<a NAME="E1:67_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">X1</font> =  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> ;<br/>
<a NAME="E3:67_1"/>
<font color="Maroon">X1</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:67_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Omega</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:67_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/><a NAME="E2:67_2"/><b>then </b>
 ex <font color="Olive">S</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> )
 ;<br/><b>hence </b><a NAME="E3:67_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:67"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">BSeq</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</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">BSeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span>  ) holds <br/> <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Olive">BSeq</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">BSeq</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>;<br/>

<b>assume </b><a NAME="E1:67_3"/><i><font color="Green">E36</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">BSeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:67_3_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/><a NAME="E2:67_3_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">S</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:67_3_1_1"/>
<font color="Maroon">BSeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="prob_1.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E2:67_3_1_1"/>
<font color="Maroon">BSeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T9" target="_self">Th9</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:67_3_1"/>
 <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">BSeq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67_3"/>
 <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">BSeq</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:67"/>
for <font color="Olive">E</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>  st <font color="Olive">E</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span>  holds <br/><font color="Olive">E</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">E</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>;<br/>

<b>assume </b><a NAME="E1:67_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:67_4_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/><a NAME="E2:67_4_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T6" target="_self">Th6</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E3:67_4_1"/>
 ex <font color="Olive">S</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> )
 <b>by </b><i><a class="ref" href="prob_1.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E4:67_4_1"/>
<font color="Maroon">E</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67_4"/>
<font color="Maroon">E</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span>  as   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> <b>by </b><i>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a>, <a class="ref" href="prob_1.html#T4" target="_self">Th4</a>, , , </i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:67_5"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 ;<br/><a NAME="E2:67_5"/><b>then </b>
 ex <font color="Olive">S</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> )
 ;<br/><b>hence </b><a NAME="E3:67_5"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E7:67"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> holds <br/><font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_6"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> ;<br/>
<a NAME="E3:67_6"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font> : ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="prob_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E4:67_6"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:67"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font> holds <br/><font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) )
 <b>by </b><i>, <a class="ref" href="prob_1.html#T6" target="_self">Th6</a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


</div>
