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<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/>
<a NAME="E2:32_3_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E3:32_3_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/>
<a NAME="E4:32_3_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:32_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:32_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="heyting2.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_3_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E3:32_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>


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