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

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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">F</font> being <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st  <a href="card_1.html#K4">card</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">F</font> &amp; ( for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/> not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) );<br/>
<a NAME="E1:1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E37</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

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

<b>thus </b><a NAME="E2:1_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/>

<b>thus </b><a NAME="E3:1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/>

<b>thus </b><a NAME="E4:1_1"/>
for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>thus </b><a NAME="E2:1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:1_2_1"/><i><font color="Green">E41</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_2_1_1"/>
(  ex <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) or for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or  not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_2_1_1_1"/>
 ex <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">g</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:1_2_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E5:1_2_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E6:1_2_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E7:1_2_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:1_2_1_1_1"/><b>then </b>
not <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">FF</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/><a NAME="E10:1_2_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">FF</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">G</font> being  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:1_2_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FF</font>
 <b>and </b><br/><a NAME="E13:1_2_1_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FF</font>
 <b>and </b><br/><a NAME="E14:1_2_1_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">G</font>
;<br/><a NAME="E15:1_2_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">FF</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E16:1_2_1_1_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:1_2_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">FF</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/><a NAME="E18:1_2_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FF</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_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="E1:1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_2_1_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_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="E1:1_2_1_1_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> or <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_2_1_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_2_1_1_1_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FF</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E19:1_2_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>thus </b><a NAME="E20:1_2_1_1_1"/>
for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_2_1_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or  not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">G</font> = <font color="Maroon">F</font>;<br/><b>thus </b><a NAME="E2:1_2_1_1_2"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/><b>thus </b><a NAME="E3:1_2_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/><b>thus </b><a NAME="E4:1_2_1_1_2"/>
for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>let </b><font color="Maroon">F</font> be  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E4:1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E5:1"/>
 ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">g</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/> not <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">g</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div>
