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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">infinite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <a href="zfmisc_1.html#K1">K10</a>(<font color="Maroon">c<sub>1</sub></font>) = <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2">proj2</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:49"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a>  holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E3:49"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:49_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:49_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_2.html#E3:49_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<a NAME="E6:49_1"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_2.html#E5:49_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:49_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:49_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2">proj2</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="funct_5.html#K2">proj2</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_2.html#E2:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topgen_2.html#E5:49_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>;<br/>
<b>hence </b><a NAME="E9:49_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="topgen_2.html#E3:49_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_2.html#E2:49"><i><font color="Green">E31</font></i></a>, <a class="ref" href="card_1.html#T28">CARD_1:28</a></i>;<br/>
<b>hence </b><a NAME="E5:49"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_4.html#T87">CARD_4:87</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <a href="zfmisc_1.html#K1">K10</a>(<font color="Maroon">c<sub>1</sub></font>) = <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:49"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E8:49"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_2.html#E7:49"><i><font color="Green">E32</font></i></a></i>;<br/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>)

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="topgen_2.html#E7:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topgen_2.html#E1:49_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>8</sub></font>)
<b>by </b><i><a class="txt" href="topgen_2.html#E7:49"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a> 

;<br/>
<a NAME="E4:49_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="fin_topo.html#T29">FIN_TOPO:29</a></i>;<br/>
<b>hence </b><a NAME="E5:49_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:49"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:49_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_2.html#E7:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topgen_2.html#E3:49_3"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:49_3"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="topgen_2.html#E7:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topgen_2.html#E3:49_3"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:49_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:49"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_2.html#E7:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topgen_2.html#E8:49"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>


</div>
