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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a>  st  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">F</font>];<br/>
<a NAME="E1:85_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:85_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:85_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:85_1"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:85_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F</font>]
 <b>by </b><i><a class="ref" href="stirl2_1.html#T17" target="_self">Th17</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:85_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:85_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E2:85_1_2"/><i><font color="Green">E63</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E3:85_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:85_1_2_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">F</font>] )
 ;<br/>

<b>set </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<a NAME="E2:85_1_2_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="stirl2_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E3:85_1_2_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>hence </b><a NAME="E4:85_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>]
 <b>by </b><i><a class="ref" href="stirl2_1.html#T19" target="_self">Th19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:85_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F</font>]
 <b>by </b><i><a class="ref" href="stirl2_1.html#T18" target="_self">Th18</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E3:85_1"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:85_1"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>,<font color="Maroon">n</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T19" target="_self">Th19</a>, <a class="ref" href="card_1.html#T74">CARD_1:74</a></i>;<br/>
<a NAME="E6:85_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<a NAME="E7:85_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E8:85_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="stirl2_1.html#E2:85_1"><i><font color="Green">E58</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:85_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F</font>]
 <b>by </b><i/>;<br/>


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