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

<b>consider </b><font color="Maroon" title="c3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:23_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E3:23_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E4:23_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
 <a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">q</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">q</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow15.html#D2" target="_self" title="YELLOW15:def.2">Def2</a></i>;<br/>
<a NAME="E5:23_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="yellow15.html#T2" target="_self" title="YELLOW15:th.2">Th2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>;<br/>
<a NAME="E6:23_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">q</font>) where <font color="Olive" title="b1">q</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="yellow15.html#E5:23_1_1"><i><font color="Green" title="E12">A2</font></i></a>);<br/></i>
<a NAME="E7:23_1_1"/>
 <a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">q</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">q</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:23_1_1_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">q</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:23_1_1_1"/><i><font color="Green" title="E15">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="yellow15.html#E4:23_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>
<a NAME="E5:23_1_1_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c5">q</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E6:23_1_1_1"/>
<font color="Maroon" title="c5">q</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>hence </b><a NAME="E7:23_1_1_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><a href="yellow15.html#K1" title="YELLOW15:func.1">MergeSequence</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">q1</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">q1</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow15.html#E3:23_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yellow15.html#E4:23_1_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="yellow15.html#E5:23_1_1_1"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:23_1_1"/>
 <a href="yellow15.html#K2" title="YELLOW15:func.2">Components</a> <font color="Maroon" title="c2">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="yellow15.html#E6:23_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>


</div>
