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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<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 ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 3 ) );<br/>
<a NAME="E1:39_1_1"/><i><font color="Green">E17</font></i>: 
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="numbers.html#K5">NAT</a>  holds <br/>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="numbers.html#K5">NAT</a>  &amp; <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="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = 2, <font color="Maroon">c<sub>5</sub></font> = 1, <font color="Maroon">c<sub>6</sub></font> = 3 as    <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:39_1_1_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:39_1_1_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_3"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:39_1_1_1_1_3"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_4"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_4"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:39_1_1_1_1_4"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E18</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>and </b><br/><a NAME="E4:39_1_1"/><i><font color="Green">E19</font></i>: 
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="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="catalg_1.html#E1:39_1_1"><i><font color="Green">E17</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="catalg_1.html#E3:39_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E6:39_1_1"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="catalg_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E7:39_1_1"/><i><font color="Green">E21</font></i>: 
the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="catalg_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E8:39_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K4" target="_self">underlay</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="catalg_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="catalg_1.html#D9" target="_self">CATALG_1:def 9</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>6</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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E6:39_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="catalg_1.html#E3:39_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_2.html#D3">FINSEQ_2:def 3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = 0;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E5:39_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:39_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:39_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="catalg_1.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E7:39_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <span class="p4">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:39_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="catalg_1.html#E6:39_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="catalg_1.html#E8:39_1_1"><i><font color="Green">E22</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E10:39_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="catalg_1.html#E7:39_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="catalg_1.html#E10:39_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3_1"><b>suppose </b></a><a NAME="E1:39_1_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_3_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> 1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="catalg_1.html#E3:39_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_2.html#D3">FINSEQ_2:def 3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = 1;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><a NAME="E5:39_1_1_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:39_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_3_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <span class="p4">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E7:39_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="catalg_1.html#E8:39_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="catalg_1.html#E3:39_1_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="catalg_1.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3_2"><b>suppose </b></a><a NAME="E1:39_1_1_3_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_3_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="catalg_1.html#E3:39_1_1_3_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_2.html#D3">FINSEQ_2:def 3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = 2;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><a NAME="E5:39_1_1_3_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:39_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_3_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K4" target="_self">underlay</a> <span class="p4">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E7:39_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="catalg_1.html#E8:39_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="catalg_1.html#E3:39_1_1_3_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="catalg_1.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
