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

<b>set </b><font color="Maroon">cA</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E1:5_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
;<br/>
<a NAME="E2:5_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:5_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:5_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> or  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E6:5_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S5">RECDEF_1:sch 5</a>(, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> implies ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  ex <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Olive">X</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> ) ) );<br/>
<a NAME="E7:5_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[1,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1_1_1"/><i><font color="Green">E50</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:5_1_1_1"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:5_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E4:5_1_1_1"/><i><font color="Green">E52</font></i>: 
<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="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><a NAME="E5:5_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E7:5_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>

<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1_1_1"/>
 <a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:5_1_1_1_1"/><b>then </b>
not  <a href="subset_1.html#K10">choose</a> <font color="Maroon">A</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"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:5_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>
<a NAME="E9:5_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></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="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:5_1_1_1"/><b>then </b>
<font color="Maroon">A</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"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<b>hence </b><a NAME="E11:5_1_1_1"/>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:5_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1_1_2"/><i><font color="Green">E55</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:5_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:5_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>]
 <b>and </b><br/><a NAME="E4:5_1_1_2"/>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:5_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>and </b><br/><a NAME="E7:5_1_1_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:5_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , , <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">ba</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:5_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E11:5_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">ba</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E12:5_1_1_2"/><i><font color="Green">E63</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> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">ba</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E13:5_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:5_1_1_2"/><i><font color="Green">E65</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ba</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E15:5_1_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<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> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:5_1_1_2"/>
<span class="p1">(<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> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E17:5_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<b>reconsider </b><font color="Maroon">XX</font> = <span class="p1">(<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> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T16">FINSET_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon">XX</font>
;<br/>

<b>thus </b><a NAME="E19:5_1_1_2"/>
<font color="Maroon">XX</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>

<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1_2_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<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> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<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> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:5_1_1_2_1"/><b>then </b>
not  <a href="subset_1.html#K10">choose</a> <span class="p1">(<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> <a href="mcart_1.html#K2">`2</a> </span>)</span> <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"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:5_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:5_1_1_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E22:5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E23:5_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">XX</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<b>hence </b><a NAME="E24:5_1_1_2"/>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">XX</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:5_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="uproots.html#S1" target="_self">UPROOTS:sch 1</a>(, );<br/></i>
<a NAME="E10:5_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:5_1_1_3"/>
<font color="Maroon">y</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">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:5_1_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:5_1_1_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</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="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E5:5_1_1_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E6:5_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , </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 <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</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="E13:5_1_1"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E14:5_1_1"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E15:5_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:5_1_1"/><i><font color="Green">E98</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E17:5_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:5_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:5_1_1_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:5_1_1_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">p</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="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E5:5_1_1_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:5_1_1_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>hence </b><a NAME="E7:5_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E19:5_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>

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

<b>thus </b><a NAME="E20:5_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> &amp; ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> or  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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="Olive">x</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> ) &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">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="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  ) )
 <b>by </b><i>, , , , </i>;<br/>


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