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<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:23_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:23_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:23_1_1"/><i><font color="Green">E11</font></i>: 
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B 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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow15.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:23_1_1"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="yellow15.html#T2" target="_self">Th2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2">Element</a> of <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> =  <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">c<sub>3</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E6:23_1_1"/><i><font color="Green">E13</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="finseq_2.html#M2">Element</a> of <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a>  </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="yellow15.html#E5:23_1_1"><i><font color="Green">E12</font></i></a>);<br/></i>
<a NAME="E7:23_1_1"/>
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is    <a href="finseq_2.html#M2">Element</a> of <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">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">c<sub>4</sub></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="E1:23_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></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="E3:23_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:23_1_1_1"/><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow15.html#E4:23_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E5:23_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E6:23_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>hence </b><a NAME="E7:23_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is    <a href="finseq_2.html#M2">Element</a> of <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow15.html#E3:23_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow15.html#E4:23_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow15.html#E5:23_1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:23_1_1"/>
 <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="yellow15.html#E6:23_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


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