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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>(  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">a<sub>1</sub></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">a<sub>1</sub></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/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:38_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:38_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:38_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_1"/>
(  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">x</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">x</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>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1"/>
 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">x</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">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> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">m</font>,<font color="Maroon">k</font> )
 ;<br/><a NAME="E4:38_1_1_1_1_1"/><i><font color="Green">E49</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/><a NAME="E5:38_1_1_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:38_1_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:38_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:38_1_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_1_1_1_1"/>
(  ex <font color="Olive">m1</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m1</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> ) or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 </span><b>by </b><i>, <a class="txt" href="heyting3.html#E2:38_1_1"><i><font color="Green">E30</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_1_1"/>
 ex <font color="Olive">m1</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">m1</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m1</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_1_1_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 ;<br/><a NAME="E4:38_1_1_1_1_1_1_1_1"/><i><font color="Green">E95</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i/>;<br/><a NAME="E5:38_1_1_1_1_1_1_1_1"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="heyting3.html#E3:38_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:38_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_1_1_1_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:38_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_1_2"/><i><font color="Green">E96</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><a NAME="E2:38_1_1_1_1_1_1_1_2"/><i><font color="Green">E114</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:38_1_1_1_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="real_2.html#T146">REAL_2:146</a></i>;<br/><a NAME="E4:38_1_1_1_1_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:38_1_1_1_1_1_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E6:38_1_1_1_1_1_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:38_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E8:38_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</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"><a NAME="E2:38_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:38_1_1_1_1_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_2_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:38_1_1_1_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:38_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1_1_1_1_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:38_1_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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><a href="mcart_1.html#K12">:]</a></span> 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="E5:38_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E6:38_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:38_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:38_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as    <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> <b>by </b><i>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E8:38_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  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">x</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">x</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/>

<b>assume </b><a NAME="E9:38_1_1"/>
(  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">x</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">x</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/>

<b>hence </b><a NAME="E10:38_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div>
