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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="hilbert3.html#NM1">SetValuation</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert3.html#D7" target="_self">HILBERT3:def 7</a><br/>







<a NAME="E1:55"/><i><font color="Green">E44</font></i>: 
(  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1">Function</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="funct_6.html#K7">Frege</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:55"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>

<a NAME="E1:55_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="fraenkel.html#M2">Element</a> of  <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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E2:55_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is    <a href="fraenkel.html#M2">Element</a> of  <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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E3:55_1"/><b>then </b>
 <a href="pralg_2.html#K3">Frege</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E1:55"><i><font color="Green">E44</font></i></a>, <a class="ref" href="hilbert3.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E4:55_1"/><b>then </b>
 <a href="pralg_2.html#K3">Frege</a> <font color="Maroon">c<sub>5</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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E5:55_1"/>
 <a href="pralg_2.html#K3">Frege</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:55"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>(<a class="txt" href="hilbert3.html#E2:55"><i><font color="Green">E45</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hilbert3.html#M1" target="_self">Permutation</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<a NAME="E5:55"/>
<font color="Maroon">c<sub>5</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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></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>4</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E6:55"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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="E7:55"/>
<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"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/><a NAME="E2:55_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:55_2"/><b>then </b>
<font color="Maroon">c<sub>7</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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E5:55_2"/><i><font color="Green">E47</font></i>: 
 <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/><b>then </b><i><font color="Green">E48</font></i>:  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T20">CARD_3:20</a></i>
;<br/><a NAME="E7:55_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</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"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E8:55_2"/>
<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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self">Th38</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="pralg_2.html#K3">Frege</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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="hilbert3.html#E1:55"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbert3.html#E5:55_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="hilbert3.html#T24" target="_self">Th24</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>;<br/><a NAME="E10:55_2"/>
 <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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> as   <a href="pboole.html#NM2">ManySortedFunction</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="txt" href="hilbert3.html#E7:55_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E12:55_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K15">..</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:55_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="hilbert3.html#E6:55_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="hilbert3.html#E1:55_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>9</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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E4:55_2_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:55_2_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<a NAME="E6:55_2_2"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E7:55_2_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>15</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2_2"><i><font color="Green">E52</font></i></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/>
<a NAME="E8:55_2_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="pralg_1.html#K15">..</a> <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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E9:55_2_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="pralg_1.html#K15">..</a> <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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="hilbert3.html#E6:55_2_2"><i><font color="Green">E53</font></i></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/>
<a NAME="E11:55_2_2"/>
<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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E12:55_2_2"/><b>then </b>
<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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E13:55_2_2"/><b>then </b>
<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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>then </b><i><font color="Green">E55</font></i>:  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <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>6</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<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>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>
;<br/>
<a NAME="E15:55_2_2"/>
<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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <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>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E16:55_2_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <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>6</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E14:55_2_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E17:55_2_2"/><b>then </b><i><font color="Green">E57</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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></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> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E14:55_2_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="hilbert3.html#T38" target="_self">Th38</a></i>;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">c<sub>8</sub></font> = 
<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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> 
<b>by </b><i><a class="ref" href="hilbert3.html#T36" target="_self">Th36</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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="hilbert3.html#K1" target="_self">=&gt;</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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T27" target="_self">Th27</a></i>
;<br/>
<i><font color="Green">E59</font></i>: <font color="Maroon">c<sub>11</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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></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>14</sub></font></span>)</span> = 
<font color="Maroon">c<sub>11</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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>15</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>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self">Th38</a></i>
<br/>.= 
<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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="pralg_1.html#K15">..</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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E16:55_2_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="pralg_2.html#D8">PRALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</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>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pralg_1.html#K15">..</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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="pralg_1.html#K15">..</a> <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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T18" target="_self">Th18</a></i>
<br/>.= 
<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>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E18:55_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="hilbert3.html#T28" target="_self">Th28</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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></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>17</sub></font>
<b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self">Th38</a></i>
;<br/>
<b>thus </b><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></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>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E7:55_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="hilbert3.html#E1:55_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></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>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E17:55_2_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="hilbert3.html#E19:55_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></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>17</sub></font>
<b>by </b><i><a class="txt" href="hilbert3.html#E9:55_2_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T88">FUNCT_2:88</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="hilbert3.html#E9:55_2_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="pralg_2.html#K3">Frege</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="hilbert3.html#E4:55"><i><font color="Green">E46</font></i></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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E12:55_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="pralg_2.html#D8">PRALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="hilbert3.html#T37" target="_self">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <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>6</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E4:55"><i><font color="Green">E46</font></i></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>6</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<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>6</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon">c<sub>5</sub></font> = 
<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>6</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</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>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hilbert1.html#K3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="hilbert3.html#T37" target="_self">Th37</a></i>
;<br/><br/>


</div>
