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<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> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:44"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> holds <br/><font color="Olive">u</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:44"/><i><font color="Green">E31</font></i>: 
 <a href="substlat.html#K1">SubstitutionSet</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:44"/>
for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>, <font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>;<br/>

<b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> )
 ;<br/>

<a NAME="E2:44_1"/><b>then </b><i><font color="Green">E48</font></i>: 
(  ex <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) or <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:44_1"/><i><font color="Green">E49</font></i>: 
(  ex <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> )
 
<b>by </b><i><a class="txt" href="heyting3.html#E2:44"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1"/>
( (  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) ) or (  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> ) or ( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) ) or ( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> ) )
 </span><b>by </b><i>, <a class="txt" href="heyting3.html#E1"><i><font color="Green">Lemma33</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1"/><i><font color="Green">E50</font></i>: 
(  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:44_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">m1</font>,<font color="Maroon">k</font> )
 ;<br/><b>consider </b><font color="Maroon">m2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:44_1_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">m2</font>,<font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:44_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:44"><i><font color="Green">E31</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_2"/>
(  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:44"><i><font color="Green">E31</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_1_3"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Olive">m</font>,<font color="Maroon">k</font> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_3"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:44"><i><font color="Green">E31</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_4"><b>suppose </b></a><a NAME="E1:44_1_1_4"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:44"/><b>then </b>
 <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> : ( ( for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">u</font> is <a href="finset_1.html#V1">finite</a> ) &amp; ( for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:44"/>
 <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="substlat.html#D1">SUBSTLAT:def 1</a></i>;<br/>


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