<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</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">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Olive">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">S</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <br/>(  <a href="zf_model.html#K2">Free</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">M</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">S</font> iff <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">H</font> ) ) );<br/>
<a NAME="E1:10"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E38</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E2:10_1"/>
(  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">M</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> iff <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> ) ) )
 <b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E2:10_2"/><i><font color="Green">E58</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:10_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="nat_1.html#T21">NAT_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">e</font> as   <a href="zf_lang.html#NM1">Variable</a> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:10_2"/><i><font color="Green">E63</font></i>: 
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T67">ZF_LANG1:67</a></i>;<br/>
<a NAME="E7:10_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:10_2"/><b>then </b>
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E9:10_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<a NAME="E10:10_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T2">XCMPLX_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">S</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E12:10_2"/><i><font color="Green">E64</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">M</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font> iff <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font> ) ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E13:10_2"/>
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">M</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E14:10_2"/>
( <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> iff <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T25">ZF_MODEL:25</a></i>;<br/>
<b>hence </b><a NAME="E15:10_2"/>
( <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font> iff <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E4:10"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:10"/>
 ex <font color="Olive">S</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <br/>(  <a href="zf_model.html#K2">Free</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">M</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">S</font> iff <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
