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

<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b2">S</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> st <br/>(  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b3">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b2">S</font> iff <font color="Olive" title="b3">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> ) ) );<br/>
<a NAME="E1:10"/><i><font color="Green" title="E9">A1</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" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>take </b>
<font color="Maroon" title="c2">H</font>
;<br/>

<b>thus </b><a NAME="E2:10_1"/>
(  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c2">H</font> iff <font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c2">H</font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:10_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10"/><i><font color="Green" title="E10">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">i</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c3">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E2:10_2"/><i><font color="Green" title="E12">A5</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<a NAME="E3:10_2"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="nat_1.html#T5" title="NAT_1:th.5">NAT_1:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">e</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:10_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:10_2"/><i><font color="Green" title="E14">A7</font></i>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T67" title="ZF_LANG1:th.67">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" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:10_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:10_2"/><b>then </b>
(  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E6:10_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">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" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<a NAME="E10:10_2"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E2:10_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T2" title="XCMPLX_1:th.2">XCMPLX_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">S</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><br/><a NAME="E12:10_2"/><i><font color="Green" title="E15">A8</font></i>: 
(  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c6">S</font> iff <font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:10_2"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">S</font>
;<br/>

<b>thus </b><a NAME="E13:10_2"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="zfrefle1.html#E12:10_2"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E14:10_2"/>
( <font color="Maroon" title="c7">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> iff <font color="Maroon" title="c7">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H</font> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T25" title="ZF_MODEL:th.25">ZF_MODEL:25</a></i>;<br/>
<b>hence </b><a NAME="E15:10_2"/>
( <font color="Maroon" title="c7">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c6">S</font> iff <font color="Maroon" title="c7">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E12:10_2"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10"/><i><font color="Green" title="E11">A9</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="zfrefle1.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E2:10"><i><font color="Green" title="E10">A3</font></i></a>);<br/></i>
<a NAME="E4:10"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c1">H</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:10"/>
 ex <font color="Olive" title="b1">S</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> st <br/>(  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b2">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">S</font> iff <font color="Olive" title="b2">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c1">H</font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:10"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>


</div>
