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

<b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F</font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E23</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="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> holds <br/> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> ;<br/>
<a NAME="E4:26"/><i><font color="Green">E25</font></i>: 
 <a href="card_1.html#K4">card</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> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/>
<b>reconsider </b><font color="Maroon">dA</font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">dA</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:26"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">dA</font>  holds <font color="Maroon">F</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:26_1"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E2:26_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>;<br/>
<a NAME="E4:26_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:26_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a></i>;<br/>
<a NAME="E6:26_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
;<br/>
<b>hence </b><a NAME="E7:26_1"/>
<font color="Maroon">F</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>)
 <b>by </b><i><a class="txt" href="hallmar1.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:26"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">dA</font>,<font color="Maroon">F</font> st <br/>for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">dA</font> holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 
<b>from </b><i><a class="ref" href="nattra_1.html#S1">NATTRA_1:sch 1</a>(<a class="txt" href="hallmar1.html#E2:26"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">dA</font>,<font color="Maroon">F</font><b> such that </b><br/><a NAME="E9:26"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">dA</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 ;<br/>
<a NAME="E10:26"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<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="E12:26"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E13:26"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E14:26"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><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="E1:26_2"/>
<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>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:26_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, </i>;<br/>
<a NAME="E5:26_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:26_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:26_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E16:26"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:26"/><i><font color="Green">E78</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="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E18:26"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><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> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 ;<br/>

<b>thus </b><a NAME="E2:26_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<b>set </b><font color="Maroon">J</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span>;<br/>
<a NAME="E2:26_3_1"/><i><font color="Green">E81</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<a NAME="E3:26_3_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:26_3_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E5:26_3_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</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="txt" href="hallmar1.html#E1:26"><i><font color="Green">E23</font></i></a>, , <a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:26_3_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T49">CARD_1:49</a></i>;<br/>
<a NAME="E8:26_3_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</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="txt" href="hallmar1.html#E1:26"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:26_3_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T49">CARD_1:49</a></i>;<br/>
<a NAME="E11:26_3_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:26_3_1_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_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="E1:26_3_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:26_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_3_1_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:26_3_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, <a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:26_3_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_3_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<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="E2:26_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:26_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, <a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:26_3_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:26_3_1_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_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> ; <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="E1:26_3_1_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:26_3_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_3_1_2_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:26_3_1_2_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, <a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:26_3_1_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_3_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="hallmar1.html#T2" target="_self">Th2</a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_3_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<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="E2:26_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:26_3_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hallmar1.html#E6:26"><i><font color="Green">E26</font></i></a>, <a class="txt" href="hallmar1.html#E9:26"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:26_3_1"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E10:26"><i><font color="Green">E32</font></i></a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i>, , <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E15:26_3_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E19:26"/><b>then </b>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font>
 
;<br/>
<a NAME="E20:26"/><b>then </b>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E21:26"/><b>then </b>
<font color="Maroon">f</font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="hallmar1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E22:26"/>
 ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">X</font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">A</font>
 ;<br/>


</div>
