<?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">Y</font> be  <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:24"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E34</font></i>: 
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<font color="Olive">q</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> </span>}</span> 
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:24"/>
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:24_1"/>
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:24_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:24_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:24_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E6:24_1_1"/><b>then </b>
 <a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:24_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E8:24_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:24_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">z</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="E2:24_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:24_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:24_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E5:24_1"/><b>then </b>
 <a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:24_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E7:24_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E8:24_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , </i>;<br/>


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


</div>
