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

<b>let </b><font color="Maroon">A</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:74"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:74"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="finseq_4.html#E15:74"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:74"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:74"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="finseq_4.html#E15:74"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E9:74"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>,<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>,<font color="Maroon">q</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/>
<a NAME="E10:74"/><b>then </b>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>,<font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>,<font color="Maroon">B</font> <a href="tarski.html#R2">are_equipotent</a>  &amp; <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="tarski.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, , <a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E11:74"/><b>then </b>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>,<font color="Maroon">B</font> <a href="tarski.html#R2">are_equipotent</a>  &amp; <font color="Maroon">B</font>, <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="tarski.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<a NAME="E12:74"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>, <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
;<br/>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_3_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_3_1_1"><b>suppose </b></a><a NAME="E1:74_3_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_3_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E3:74_3_1_1_1"/>
<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> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:74_3_1_1_1"/><b>then </b>
 <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="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E5:74_3_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, , <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><b>hence </b><a NAME="E6:74_3_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:74_3_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_3_1_2"><b>suppose </b></a><a NAME="E1:74_3_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:74_3_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E3:74_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E4:74_3_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E5:74_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_4.html#E1:74_3_1_2"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E73</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>then reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E8:74_3_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/><a NAME="E9:74_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/><a NAME="E10:74_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E11:74_3_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:74_3_1_2"/>
<font color="Maroon">q</font> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/><a NAME="E13:74_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <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#T46">FUNCT_1:46</a></i>;<br/><a NAME="E14:74_3_1_2"/><b>then </b>
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/><a NAME="E15:74_3_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/><a NAME="E16:74_3_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><i><font color="Green">E77</font></i>: <font color="Maroon">q</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> = 
<font color="Maroon">q</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
;<br/><a NAME="E18:74_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:74_3_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/><a NAME="E20:74_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E1:74_3_1_2"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E21:74_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" target="_self">Def4</a>, , <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:74"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>


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