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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 );<br/>
<a NAME="E1:71"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E3:71_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K4">INT</a> , <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:71"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="hilbert3.html#E1:71"><i><font color="Green">E50</font></i></a>);<br/></i>
<a NAME="E4:71"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:71"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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>assume </b><a NAME="E1:71_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <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">c<sub>6</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:71_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="txt" href="hilbert3.html#E3:71"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:71_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="txt" href="hilbert3.html#E3:71"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:71_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E8:71_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E4:71_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="hilbert3.html#E6:71_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T2">XCMPLX_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:71"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  iff ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E4:71_3_1"/>
<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>1</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E4:71"><i><font color="Green">E52</font></i></a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E5:71_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</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="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="txt" href="hilbert3.html#E3:71"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hilbert3.html#E4:71"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E6:71_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T27">XCMPLX_1:27</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:71_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:71_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:71_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="hilbert3.html#E3:71"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hilbert3.html#E4:71"><i><font color="Green">E52</font></i></a>, <a class="txt" href="hilbert3.html#E2:71_3"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:71_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="txt" href="hilbert3.html#E3:71_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:71"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="txt" href="hilbert3.html#E5:71"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0;<br/>
<a NAME="E9:71"/><i><font color="Green">E55</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/>
<a NAME="E10:71"/>
0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0 is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <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>assume </b><a NAME="E1:71_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> )
 ;<br/>

<a NAME="E2:71_4"/><b>then </b><i><font color="Green">E56</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="hilbert3.html#E9:71"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:71_4"/>
( <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/>
<b>hence </b><a NAME="E4:71_4"/>
(  not <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="hilbert3.html#E2:71_4"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> 1,0 as   <a href="funct_2.html#NM2">Permutation</a> of 2 <b>by </b><i><a class="txt" href="hilbert3.html#E9:71"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E14:71"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_6"><b>proof </b></a><div class="add">

<a NAME="E1:71_6"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a>, <a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E2:71_6"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<b>assume </b><a NAME="E3:71_6"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a><br/>

<a NAME="E4:71_6"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> or 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="hilbert3.html#E2:71_6"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_6"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="hilbert3.html#NM1">SetValuation</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<i><font color="Green">E56</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<a NAME="E17:71"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_8_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_8_1_1"><b>suppose </b></a><a NAME="E1:71_8_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:71_8_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</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/><a NAME="E3:71_8_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="hilbert3.html#E1:71_8_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><a NAME="E5:71_8_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="hilbert3.html#E2:71_8_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="hilbert3.html#E1:71_8_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E7:71_8_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E4:71_8_1_1"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_8_1_2"><b>suppose </b></a><a NAME="E1:71_8_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:71_8_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</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/><a NAME="E3:71_8_1_2"/><b>then </b>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E5:71_8_1_2"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="hilbert3.html#E2:71_8_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="numbers.html#K4">INT</a> 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E7:71_8_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E4:71_8_1_2"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as    <a href="hilbert3.html#M1" target="_self">Permutation</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="hilbert3.html#E16:71"><i><font color="Green">E56</font></i></a>, <a class="ref" href="hilbert3.html#D4" target="_self">Def4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> or ( <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>;<br/>
<a NAME="E19:71"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> or ( <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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="E1:71_9"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> or ( <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:71_9"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:71_9"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> or ( <font color="Maroon">c<sub>13</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> ) )
 ;<br/>
<b>thus </b><a NAME="E5:71_9"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hilbert3.html#E3:71_9"><i><font color="Green">E57</font></i></a>, <a class="ref" href="hilbert3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> or ( <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E21:71"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E22:71"/><b>then </b><i><font color="Green">E58</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E59</font></i>: 2 = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="hilbert3.html#E22:71"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T30" target="_self">Th30</a></i>
;<br/>
<a NAME="E24:71"/><i><font color="Green">E60</font></i>: 
not 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E21:71"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E61</font></i>:  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="hilbert3.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="hilbert3.html#E24:71"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="numbers.html#K4">INT</a> 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<i><font color="Green">E62</font></i>:  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">2 <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a> </span>)</span>
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="funct_4.html#T68">FUNCT_4:68</a></i>
<br/>.= 
 <a href="fraenkel.html#K1">Funcs</a> 2,<a href="numbers.html#K4">INT</a> 
<b>by </b><i><a class="ref" href="card_3.html#T20">CARD_3:20</a></i>
<br/>.= 
 <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E23:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="hilbert3.html#E25:71"><i><font color="Green">E61</font></i></a>, <a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <b>by </b><i><a class="txt" href="hilbert3.html#E23:71"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E28:71"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E29:71"/><b>then </b><i><font color="Green">E63</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E64</font></i>:  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="hilbert3.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="hilbert3.html#E29:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E31:71"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_14"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span>;<br/>
<a NAME="E1:71_14"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E2:71_14"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:71_14"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D3">ABIAN:def 3</a><br/>

 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> = 
 <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"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E26:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>then </b><i><font color="Green">E66</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E26:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>
;<br/>
<a NAME="E6:71_14"/>
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_14_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:71_14_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:71_14_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E3:71_14_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E26:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E6:71_14_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E23:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="hilbert3.html#E25:71"><i><font color="Green">E61</font></i></a>, <a class="ref" href="hilbert3.html#T12" target="_self">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:71_14_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E9:71_14_3"/><b>then </b><i><font color="Green">E69</font></i>: 
not 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E70</font></i>:  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="hilbert3.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="hilbert3.html#E9:71_14_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E11:71_14_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E4:71"><i><font color="Green">E52</font></i></a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E12:71_14_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="hilbert3.html#E3:71"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hilbert3.html#E4:71"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E13:71_14_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T27">XCMPLX_1:27</a></i>;<br/>
<a NAME="E14:71_14_3"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="hilbert3.html#E11:71_14_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<a NAME="E15:71_14_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E4:71"><i><font color="Green">E52</font></i></a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E16:71_14_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="hilbert3.html#E3:71"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hilbert3.html#E4:71"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E17:71_14_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T27">XCMPLX_1:27</a></i>;<br/>
<a NAME="E18:71_14_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="txt" href="hilbert3.html#E7:71"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<b>then </b><i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>16</sub></font> = 
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E6:71_14_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="hilbert3.html#E10:71_14_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="hilbert3.html#T14" target="_self">Th14</a></i>
<br/>.= 
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E14:71_14_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="hilbert3.html#E15:71_14_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="hilbert3.html#E17:71_14_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>
<i><font color="Green">E76</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E5:71_14"><i><font color="Green">E66</font></i></a>, <a class="txt" href="hilbert3.html#E1:71_14_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E30:71"><i><font color="Green">E64</font></i></a>, <a class="txt" href="hilbert3.html#E19:71_14_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="hilbert3.html#T13" target="_self">Th13</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_14_3_4"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> or  not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_1"><b>suppose </b></a><a NAME="E1:71_14_3_4_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:71_14_3_4_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E4:71_14_3_4_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>22</sub></font> or ( <font color="Maroon">c<sub>21</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> ) )
 ;<br/><a NAME="E5:71_14_3_4_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="hilbert3.html#E6:71_14_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="hilbert3.html#E3:71_14_3_4_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="hilbert3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E6:71_14_3_4_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:71_14_3_4_1"><i><font color="Green">E77</font></i></a></i>;<br/><a NAME="E7:71_14_3_4_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><a NAME="E8:71_14_3_4_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E9:71_14_3_4_1"/><b>then </b><i><font color="Green">E82</font></i>: 
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#T11" target="_self">Th11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_14_3_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_14_3_4_1_1"/>
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:71_14_3_4_1_1"/><i><font color="Green">E83</font></i>: 
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E4:71_14_3_4_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>24</sub></font> or ( <font color="Maroon">c<sub>23</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) )
 ;<br/><a NAME="E5:71_14_3_4_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="txt" href="hilbert3.html#E3:71_14_3_4_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="hilbert3.html#T10" target="_self">Th10</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_14_3_4_1_1_1"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>24</sub></font> or ( <font color="Maroon">c<sub>23</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="hilbert3.html#E4:71_14_3_4_1_1"><i><font color="Green">E84</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:71_14_3_4_1_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:71_14_3_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbert3.html#E4:71_14_3_4_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="hilbert3.html#E5:71_14_3_4_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="hilbert3.html#E5:71_14_3_4_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:71_14_3_4_1_1_1_2"/>
( <font color="Maroon">c<sub>23</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:71_14_3_4_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbert3.html#E4:71_14_3_4_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="hilbert3.html#E5:71_14_3_4_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="hilbert3.html#E5:71_14_3_4_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="hilbert3.html#T2" target="_self">Th2</a>, <a class="ref" href="xcmplx_1.html#T19">XCMPLX_1:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E11:71_14_3_4_1"/><b>then </b>
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E9:71_14_3_4_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
0
<b>by </b><i><a class="txt" href="hilbert3.html#E20:71_14_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_3.html#T43">FUNCT_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="hilbert3.html#E7:71_14_3_4_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_3.html#T43">FUNCT_3:43</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_2"><b>suppose </b></a><a NAME="E1:71_14_3_4_2"/><i><font color="Green">E77</font></i>: 
not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:71_14_3_4_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hilbert3.html#E6:71_14_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="hilbert3.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:71_14_3_4_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:71_14_3_4_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:71_14_3_4_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E5:71_14_3_4_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; (  not <font color="Maroon">c<sub>19</sub></font> is <a href="abian.html#V1">even</a> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> ) )
 <b>by </b><i><a class="txt" href="hilbert3.html#E6:71_14_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="hilbert3.html#E1:71_14_3_4_2"><i><font color="Green">E77</font></i></a></i>;<br/><a NAME="E6:71_14_3_4_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_14_3_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_14_3_4_2_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>20</sub></font> or ( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; not <font color="Maroon">c<sub>19</sub></font> is <a href="abian.html#V1">even</a> ) )
 </span><b>by </b><i><a class="txt" href="hilbert3.html#E5:71_14_3_4_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:71_14_3_4_2_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><a NAME="E2:71_14_3_4_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E3:71_14_3_4_2_1_1_1"/>
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E6:71_14_3_4_2"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_14_3_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:71_14_3_4_2_1_1_2"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; not <font color="Maroon">c<sub>19</sub></font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><a NAME="E2:71_14_3_4_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 is <a href="abian.html#V1">even</a> )
 <b>by </b><i><a class="ref" href="hilbert3.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E3:71_14_3_4_2_1_1_2"/>
0,1 <a href="funct_4.html#K4">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E6:71_14_3_4_2"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
1
<b>by </b><i><a class="txt" href="hilbert3.html#E20:71_14_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="hilbert3.html#E4:71_14_3_4_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:71_14"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E5:71_14"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E30:71"><i><font color="Green">E64</font></i></a>, <a class="ref" href="hilbert3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </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="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="hilbert3.html#T37" target="_self">Th37</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E32:71"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>1</sub></font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:71_15"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D3">ABIAN:def 3</a><br/>

<a NAME="E2:71_15"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:71_15"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i><a class="ref" href="hilbert3.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E4:71_15"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:71_15"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:71_15"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> 2</span>)</span> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<a NAME="E7:71_15"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<b>assume </b><a NAME="E8:71_15"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_15_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="hilbert3.html#E3:71_15"><i><font color="Green">E66</font></i></a>, <a class="txt" href="hilbert3.html#E7:71_15"><i><font color="Green">E67</font></i></a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_15_1_1"><b>suppose </b></a><a NAME="E1:71_15_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:71_15_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbert3.html#E30:71"><i><font color="Green">E64</font></i></a>, <a class="txt" href="hilbert3.html#E8:71_15"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_15_1_2"><b>suppose </b></a><a NAME="E1:71_15_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:71_15_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbert3.html#E30:71"><i><font color="Green">E64</font></i></a>, <a class="txt" href="hilbert3.html#E8:71_15"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E33:71"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><a href="hilbert2.html#K1">prop</a> 1</span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> 0</span>)</span> is <a href="hilbert3.html#V2" target="_self">pseudo-canonical</a>
 <b>by </b><i><a class="txt" href="hilbert3.html#E31:71"><i><font color="Green">E65</font></i></a>, <a class="ref" href="hilbert3.html#T54" target="_self">Th54</a></i>;<br/>


</div>
