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

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Pn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E1:39_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<a NAME="E2:39_1_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>
 
;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:39_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y0</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<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> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) );<br/>
<a NAME="E4:39_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_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_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="hilbert2.html#E4:39_1"><i><font color="Green">E37</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_2_1_1"><b>suppose </b></a><a NAME="E1:39_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:39_1_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font>
 ;<br/><b>take </b>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>)
;<br/><b>thus </b><a NAME="E4:39_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  implies <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:39_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbert2.html#E2:39"><i><font color="Green">E30</font></i></a></i>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:39_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">k</font>
 ;<br/><b>hence </b><a NAME="E6:39_1_2_1_1"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">k</font>)
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:39_1_2"><i><font color="Green">E38</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_2"><b>suppose </b></a><a NAME="E1:39_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">F<sub>1</sub></font>()
;<br/><b>thus </b><a NAME="E3:39_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">F<sub>1</sub></font>()]
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:39_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbert2.html#E2:39"><i><font color="Green">E30</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">M0</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E6:39_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="hilbert2.html#E1:39_1"><i><font color="Green">E36</font></i></a>);<br/></i>
<a NAME="E7:39_1"/>
 <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:39_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:39_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="hilbert2.html#E4:39_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E10:39_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:39_1_3"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E2:39_1_3"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:39_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:39_1_4"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:39_1_4"/>
<font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i><a class="txt" href="hilbert2.html#E4:39_1"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:39_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds <br/> ( not <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> &amp; not <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_5"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_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_5_1"/>
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  )
 </span><b>by </b><i>, <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_5_1_1"><b>suppose </b></a><a NAME="E1:39_1_5_1_1"/>
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:39_1_5_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:39_1_5_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_2"><b>suppose </b></a><a NAME="E1:39_1_5_1_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:39_1_5_1_2"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>
 ;<br/><b>hence </b><a NAME="E3:39_1_5_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="hilbert2.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_3"><b>suppose </b></a><a NAME="E1:39_1_5_1_3"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:39_1_5_1_3"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>
 ;<br/><b>hence </b><a NAME="E3:39_1_5_1_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:39_1"/><b>then </b><i><font color="Green">E90</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> )
 
;<br/>
<a NAME="E14:39_1"/>
( ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 
<b>by </b><i><a class="ref" href="hilbert2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E15:39_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 
<b>by </b><i>, , , <a class="txt" href="hilbert2.html#E6:39_1"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E16:39_1"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span>  &amp; <font color="Olive">Z</font> is <a href="ordinal1.html#V6">c=-linear</a> holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:39_1_6"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:39_1_6"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:39_1_6"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<a NAME="E4:39_1_6"/><i><font color="Green">E95</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="hilbert1.html#K1">HP-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 ;<br/>

<a NAME="E2:39_1_6_1"/><b>then </b>
 ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:39_1_6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="hilbert1.html#K1">HP-WFF</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:39_1_6"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span>  is   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="hilbert1.html#K1">HP-WFF</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">uZ</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i>, <a class="ref" href="borsuk_1.html#T25">BORSUK_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon">Z0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:39_1_6"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="hilbert2.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E9:39_1_6"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">Z0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">uZ</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E10:39_1_6"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<a NAME="E1:39_1_6_2"/>
<font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:39_1_6_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z0</font>
 <b>and </b><br/><a NAME="E4:39_1_6_2"/><i><font color="Green">E100</font></i>: 
 <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:39_1_6_2"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:39_1_6_2"/>
( ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) ) )
 ;<br/>
<b>assume </b><a NAME="E7:39_1_6_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_2_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_6_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  )
 </span><b>by </b><i>, <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_6_2_1_1"><b>suppose </b></a><a NAME="E1:39_1_6_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="hilbert1.html#K2">VERUM</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_6_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_2_1_2"><b>suppose </b></a><a NAME="E1:39_1_6_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:39_1_6_2_1_2"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>
 ;<br/><b>hence </b><a NAME="E3:39_1_6_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:39_1_6"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">uZ</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:39_1_6"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:39_1_6_3"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:39_1_6_3"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E13:39_1_6"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_6_4"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_4_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_6_4_1"/>
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_4_1_1"><b>suppose </b></a><a NAME="E1:39_1_6_4_1_1"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Z0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_6_4_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font>
 <b>and </b><br/><a NAME="E4:39_1_6_4_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:39_1_6_4_1_1"/>
<font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:39_1_6_4_1_1"/><b>then </b>
 ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Maroon">Z0</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) ) )
 ;<br/><a NAME="E7:39_1_6_4_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:39_1_6_4_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_4_1_2"><b>suppose </b></a><a NAME="E1:39_1_6_4_1_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Z0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_6_4_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font>
 <b>and </b><br/><a NAME="E4:39_1_6_4_1_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:39_1_6_4_1_2"/>
<font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:39_1_6_4_1_2"/><b>then </b>
 ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Maroon">Z0</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) ) )
 ;<br/><a NAME="E7:39_1_6_4_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:39_1_6_4_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">a<sub>1</sub></font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) );<br/>
<a NAME="E14:39_1_6"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_1_6_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:39_1_6_5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E4:39_1_6_5"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:39_1_6_5"/>
(  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) ) )
 <b>and </b><br/><a NAME="E6:39_1_6_5"/><i><font color="Green">E137</font></i>: 
 ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 ;<br/>
<b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:39_1_6_5"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E9:39_1_6_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">M</font>]
 <b>by </b><i>, <a class="ref" href="hilbert2.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">MM</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E16:39_1_6"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E17:39_1_6"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">MM</font> is <a href="fraenkel.html#V1">functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a><br/>

<b>assume </b><a NAME="E1:39_1_6_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">MM</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_6_6"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 <b>and </b><br/><a NAME="E4:39_1_6_6"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:39_1_6_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:39_1_6_6"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 
<b>by </b><i>, <a class="ref" href="hilbert2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E7:39_1_6_6"/>
<font color="Maroon">y</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">rr</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">MM</font> as  <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E19:39_1_6"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> holds <br/><font color="Olive">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:39_1_6_7"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_6_7"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 <b>and </b><br/><a NAME="E4:39_1_6_7"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:39_1_6_7"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T5" target="_self">Th5</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:39_1_6_7"/><b>then </b><i><font color="Green">E149</font></i>: 
 ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">x1</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E7:39_1_6_7"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:39_1_6_7"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 <b>and </b><br/><a NAME="E10:39_1_6_7"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:39_1_6_7"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:39_1_6_7"/><b>then </b><i><font color="Green">E153</font></i>: 
 ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">x2</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E13:39_1_6_7"/><i><font color="Green">E154</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E14:39_1_6_7"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E15:39_1_6_7"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E16:39_1_6_7"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T7" target="_self">Th7</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E17:39_1_6_7"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E18:39_1_6_7"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
<b>by </b><i>, <a class="ref" href="hilbert2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E19:39_1_6_7"/><i><font color="Green">E160</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_6_7_1"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font>
 ;<br/>

<b>thus </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbert2.html#T6" target="_self">Th6</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E20:39_1_6_7"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] holds <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">s</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39_1_6_7_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> )
 <b>and </b><br/><a NAME="E2:39_1_6_7_2"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/>

<b>consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E4:39_1_6_7_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E5:39_1_6_7_2"/>
(  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 <b>and </b><br/><a NAME="E6:39_1_6_7_2"/><i><font color="Green">E165</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E7:39_1_6_7_2"/>
 ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 <b>by </b><i/>;<br/>
<a NAME="E8:39_1_6_7_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T6" target="_self">Th6</a>, <a class="ref" href="hilbert2.html#T7" target="_self">Th7</a>, <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E9:39_1_6_7_2"/>
( <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_7_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_6_7_2_1"/>
<font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font>
 ;<br/>

<a NAME="E2:39_1_6_7_2_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span>] )
 
<b>by </b><i>, <a class="ref" href="hilbert2.html#T6" target="_self">Th6</a>, <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a>, <a class="ref" href="hilbert2.html#D8" target="_self">Def8</a>, , <a class="txt" href="hilbert2.html#E8:39_1"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_6_7_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E3:39"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:39_1_6_7_2"/>
( <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_7_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_6_7_2_2"/>
<font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font>
 ;<br/>

<a NAME="E2:39_1_6_7_2_2"/><b>then </b>
( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span>] )
 
<b>by </b><i>, <a class="ref" href="hilbert2.html#T6" target="_self">Th6</a>, <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a>, <a class="ref" href="hilbert2.html#D8" target="_self">Def8</a>, , <a class="txt" href="hilbert2.html#E8:39_1"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_6_7_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert2.html#E4:39"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E21:39_1_6_7"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" target="_self">HILBERT2:sch 2</a>(, , );<br/></i>
<b>hence </b><a NAME="E22:39_1_6_7"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:39_1_6"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font> holds <br/><font color="Olive">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:39_1_6_8"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_6_8"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 <b>and </b><br/><a NAME="E4:39_1_6_8"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:39_1_6_8"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:39_1_6_8"/><b>then </b><i><font color="Green">E171</font></i>: 
 ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">x1</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E7:39_1_6_8"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:39_1_6_8"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 <b>and </b><br/><a NAME="E10:39_1_6_8"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:39_1_6_8"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T6" target="_self">Th6</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:39_1_6_8"/><b>then </b>
 ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">x2</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 
<b>by </b><i>, <a class="ref" href="hilbert2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E13:39_1_6_8"/><b>then </b><i><font color="Green">E176</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E14:39_1_6_8"/>
<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i>, , , <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E15:39_1_6_8"/><b>then </b>
( <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x2</font> or <font color="Maroon">x2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E16:39_1_6_8"/>
<font color="Maroon">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="hilbert2.html#T5" target="_self">Th5</a>, , <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:39_1_6"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">rr</font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T2">WELLFND1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">MM</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E23:39_1_6"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_9"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_1_6_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:39_1_6_9"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:39_1_6_9"/>
<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E24:39_1_6"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">uZ</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">uZ</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E26:39_1_6"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">rr</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E27:39_1_6"/>
<font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E28:39_1_6"/><b>then </b>
<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E29:39_1_6"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">MZ0</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Z0</font><b> such that </b><br/><a NAME="E31:39_1_6"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">MZ0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z0</font>
 <b>and </b><br/><a NAME="E32:39_1_6"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E33:39_1_6"/><i><font color="Green">E182</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>and </b><br/><a NAME="E34:39_1_6"/>
( ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z0</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E35:39_1_6"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MZ0</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E36:39_1_6"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, , , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E37:39_1_6"/><i><font color="Green">E185</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:39_1_6_10"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">MZ0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>)
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E38:39_1_6"/><i><font color="Green">E186</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_6_11"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_6_11"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E4:39_1_6_11"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">MZ1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Z1</font><b> such that </b><br/><a NAME="E6:39_1_6_11"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">MZ1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E7:39_1_6_11"/>
( <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) )
 <b>and </b><br/><a NAME="E8:39_1_6_11"/><i><font color="Green">E192</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <b>and </b><br/><a NAME="E9:39_1_6_11"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:39_1_6_11"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:39_1_6_11"/><b>then </b>
<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:39_1_6_11"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E13:39_1_6_11"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/>

<a NAME="E14:39_1_6_11"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:39_1_6_11"/><b>then </b>
 ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) ) )
 
;<br/>
<a NAME="E16:39_1_6_11"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 
<b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E17:39_1_6_11"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MZ1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MZ1</font> &amp; <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MZ1</font> )
 
<b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E18:39_1_6_11"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E19:39_1_6_11"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:39_1_6"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_6_12"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">uZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_6_12"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E4:39_1_6_12"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">MZ1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Z1</font><b> such that </b><br/><a NAME="E6:39_1_6_12"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">MZ1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E7:39_1_6_12"/>
( <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) )
 <b>and </b><br/><a NAME="E8:39_1_6_12"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <b>and </b><br/><a NAME="E9:39_1_6_12"/><i><font color="Green">E198</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:39_1_6_12"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MM</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:39_1_6_12"/><b>then </b>
<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rr</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:39_1_6_12"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">MM</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E13:39_1_6_12"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/>

<a NAME="E14:39_1_6_12"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:39_1_6_12"/><b>then </b>
 ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  st <br/>( <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) ) )
 
;<br/>
<a NAME="E16:39_1_6_12"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 
<b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E17:39_1_6_12"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MZ1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MZ1</font> &amp; <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MZ1</font> )
 
<b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E18:39_1_6_12"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MZ1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E19:39_1_6_12"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E40:39_1_6"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 <b>by </b><i>, , , , <a class="ref" href="hilbert2.html#T6" target="_self">Th6</a>, <a class="ref" href="hilbert2.html#T7" target="_self">Th7</a>, </i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">H</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:39_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E19:39_1"/><i><font color="Green">E203</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span>  &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> holds <br/> not <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 <b>by </b><i><a class="ref" href="hilbert2.html#T3" target="_self">Th3</a>, <a class="ref" href="orders_1.html#T177">ORDERS_1:177</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E21:39_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E22:39_1"/><i><font color="Green">E205</font></i>: 
(  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 <b>and </b><br/><a NAME="E23:39_1"/><i><font color="Green">E206</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E24:39_1"/><i><font color="Green">E207</font></i>: 
 ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 <b>by </b><i><a class="ref" href="hilbert2.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E26:39_1"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) )
 <b>and </b><br/><a NAME="E27:39_1"/><i><font color="Green">E209</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <b>and </b><br/><a NAME="E28:39_1"/><i><font color="Green">E210</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <b>by </b><i/>;<br/>
<a NAME="E29:39_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E1:39_1_7"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E2:39_1_7"/><i><font color="Green">E213</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[ <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:39_1_7"/><i><font color="Green">E214</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>] holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">s</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_7_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>assume </b><a NAME="E2:39_1_7_1"/><i><font color="Green">E216</font></i>: 
(  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>] or  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>] )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_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_7_1_1"/>
(  not <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  not <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_1"/><i><font color="Green">E217</font></i>: 
not <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E4:39_1_7_1_1_1"/>
<font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:39_1_7_1_1_1"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:39_1_7_1_1_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><div><i><font color="Green">E221</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:39_1_7_1_1_1_1"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:39_1_7_1_1_1_1"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E8:39_1_7_1_1_1"/><i><font color="Green">E222</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_7_1_1_1_2"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1_2_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_7_1_1_1_2_1"/>
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> or ( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> ) or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 </span><b>by </b><i><a class="ref" href="hilbert2.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_1_2_1_1"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:39_1_7_1_1_1_2_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <b>by </b><i><a class="ref" href="hilbert2.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1_2_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:39_1_7_1_1_1_2_1_2"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 <b>and </b><br/><a NAME="E2:39_1_7_1_1_1_2_1_2"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E3:39_1_7_1_1_1_2_1_2"/>
not <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:39_1_7_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:39_1_7_1_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:39_1_7_1_1_1_2_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_1_2_1_3"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_1_2_1_3"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E3:39_1_7_1_1_1_2_1_3"/><b>then </b>
not <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:39_1_7_1_1_1_2_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:39_1_7_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:39_1_7_1_1_1_2_1_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">CMrMs</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:39_1_7_1_1_1"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">CMrMs</font>]
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">N</font> = <font color="Maroon">M</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">CMrMs</font></span>)</span>;<br/><a NAME="E11:39_1_7_1_1_1"/><i><font color="Green">E230</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">CMrMs</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E12:39_1_7_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E13:39_1_7_1_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">CMrMs</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y'</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">N</font> = <font color="Maroon">M</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">CMrMs</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y'</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E15:39_1_7_1_1_1"/>
 <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E16:39_1_7_1_1_1"/><b>then </b>
not  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:39_1_7_1_1_1"/><b>then </b><i><font color="Green">E231</font></i>: 
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E18:39_1_7_1_1_1"/><i><font color="Green">E232</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:39_1_7_1_1_1_3"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="hilbert2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:39_1_7_1_1_1_3"/><b>then </b>
not  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>)
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div><a NAME="E19:39_1_7_1_1_1"/><i><font color="Green">E233</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_7_1_1_1_4"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:39_1_7_1_1_1_4"/><i><font color="Green">E235</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1_4_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_7_1_1_1_4_1"/>
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> or <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_1_4_1_1"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/><a NAME="E3:39_1_7_1_1_1_4_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/><a NAME="E4:39_1_7_1_1_1_4_1_1"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:39_1_7_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E238</font></i>: 
( <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E6:39_1_7_1_1_1_4_1_1"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:39_1_7_1_1_1_4_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">CMrMs</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E8:39_1_7_1_1_1_4_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i><a class="ref" href="hilbert2.html#T7" target="_self">Th7</a>, , <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_1_4_1_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E239</font></i>: 
not <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:39_1_7_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E240</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:39_1_7_1_1_1_4_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><a NAME="E5:39_1_7_1_1_1_4_1_2"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:39_1_7_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E241</font></i>: 
( <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E7:39_1_7_1_1_1_4_1_2"/>
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E8:39_1_7_1_1_1_4_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E20:39_1_7_1_1_1"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_7_1_1_1_5"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:39_1_7_1_1_1_5"/><i><font color="Green">E243</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/>

<a NAME="E3:39_1_7_1_1_1_5"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:39_1_7_1_1_1_5"/><b>then </b><i><font color="Green">E244</font></i>: 
not <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:39_1_7_1_1_1_5"/><b>then </b><i><font color="Green">E245</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:39_1_7_1_1_1_5"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:39_1_7_1_1_1_5"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:39_1_7_1_1_1_5"/><b>then </b><i><font color="Green">E246</font></i>: 
( <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<a NAME="E9:39_1_7_1_1_1_5"/>
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_7_1_1_1_5"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:39_1_7_1_1_1"/><b>then </b>
<font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 <b>by </b><i>, , , <a class="ref" href="hilbert2.html#T6" target="_self">Th6</a>, <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, , </i>;<br/><b>hence </b><a NAME="E22:39_1_7_1_1_1"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_2"/><i><font color="Green">E247</font></i>: 
not <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E4:39_1_7_1_1_2"/>
<font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:39_1_7_1_1_2"/><b>then </b><i><font color="Green">E248</font></i>: 
<font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:39_1_7_1_1_2"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><div><i><font color="Green">E250</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:39_1_7_1_1_2_1"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:39_1_7_1_1_2_1"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E8:39_1_7_1_1_2"/><i><font color="Green">E251</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_7_1_1_2_2"/><i><font color="Green">E252</font></i>: 
( <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2_2_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_7_1_1_2_2_1"/>
( <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> or ( <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> ) or <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 </span><b>by </b><i><a class="ref" href="hilbert2.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_2_2_1_1"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:39_1_7_1_1_2_2_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <b>by </b><i><a class="ref" href="hilbert2.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2_2_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:39_1_7_1_1_2_2_1_2"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 <b>and </b><br/><a NAME="E2:39_1_7_1_1_2_2_1_2"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E3:39_1_7_1_1_2_2_1_2"/>
not <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:39_1_7_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:39_1_7_1_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:39_1_7_1_1_2_2_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_2_2_1_3"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_2_2_1_3"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E3:39_1_7_1_1_2_2_1_3"/><b>then </b>
not <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:39_1_7_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:39_1_7_1_1_2_2_1_3"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:39_1_7_1_1_2_2_1_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">IMrMs</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:39_1_7_1_1_2"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">IMrMs</font>]
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">N</font> = <font color="Maroon">M</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">IMrMs</font></span>)</span>;<br/><a NAME="E11:39_1_7_1_1_2"/><i><font color="Green">E258</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">IMrMs</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E12:39_1_7_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E13:39_1_7_1_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">IMrMs</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y'</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">N</font> = <font color="Maroon">M</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">IMrMs</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Y'</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E15:39_1_7_1_1_2"/>
 <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E16:39_1_7_1_1_2"/><b>then </b>
not  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:39_1_7_1_1_2"/><b>then </b><i><font color="Green">E259</font></i>: 
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E18:39_1_7_1_1_2"/><i><font color="Green">E260</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:39_1_7_1_1_2_3"/>
 <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:39_1_7_1_1_2_3"/><b>then </b>
not  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>)
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div><a NAME="E19:39_1_7_1_1_2"/><i><font color="Green">E261</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_7_1_1_2_4"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:39_1_7_1_1_2_4"/><i><font color="Green">E263</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2_4_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_7_1_1_2_4_1"/>
( <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_2_4_1_1"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_2_4_1_1"/><b>then </b><i><font color="Green">E265</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/><a NAME="E3:39_1_7_1_1_2_4_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:39"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:39_1_7_1_1_2_4_1_1"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:39_1_7_1_1_2_4_1_1"/><b>then </b><i><font color="Green">E266</font></i>: 
( <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E6:39_1_7_1_1_2_4_1_1"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:39_1_7_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">IMrMs</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E8:39_1_7_1_1_2_4_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i><a class="ref" href="hilbert2.html#T7" target="_self">Th7</a>, , <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_7_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:39_1_7_1_1_2_4_1_2"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:39_1_7_1_1_2_4_1_2"/><b>then </b><i><font color="Green">E267</font></i>: 
not <font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:39_1_7_1_1_2_4_1_2"/><b>then </b><i><font color="Green">E268</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:39_1_7_1_1_2_4_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><a NAME="E5:39_1_7_1_1_2_4_1_2"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:39_1_7_1_1_2_4_1_2"/><b>then </b><i><font color="Green">E269</font></i>: 
( <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E7:39_1_7_1_1_2_4_1_2"/>
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E8:39_1_7_1_1_2_4_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E20:39_1_7_1_1_2"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_7_1_1_2_5"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:39_1_7_1_1_2_5"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/>

<a NAME="E3:39_1_7_1_1_2_5"/>
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:39_1_7_1_1_2_5"/><b>then </b><i><font color="Green">E272</font></i>: 
not <font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:39_1_7_1_1_2_5"/><b>then </b><i><font color="Green">E273</font></i>: 
<font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:39_1_7_1_1_2_5"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:39_1_7_1_1_2_5"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:39_1_7_1_1_2_5"/><b>then </b><i><font color="Green">E274</font></i>: 
( <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<a NAME="E9:39_1_7_1_1_2_5"/>
<font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_7_1_1_2_5"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , <a class="ref" href="hilbert2.html#D6" target="_self">Def6</a>, <a class="ref" href="hilbert2.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:39_1_7_1_1_2"/><b>then </b>
<font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="hilbert1.html#K1">HP-WFF</a>  : (  <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st ( <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> or <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) ) &amp;  ex <font color="Olive">M</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">Y</font> st <br/>( <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] ) ) ) </span>}</span> 
 <b>by </b><i>, , , <a class="ref" href="hilbert2.html#T6" target="_self">Th6</a>, <a class="ref" href="hilbert2.html#T8" target="_self">Th8</a>, , </i>;<br/><b>hence </b><a NAME="E22:39_1_7_1_1_2"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:39_1_7"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" target="_self">HILBERT2:sch 2</a>(, , );<br/></i>
<b>hence </b><a NAME="E5:39_1_7"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T49">SUBSET_1:49</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  ;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E31:39_1"/>
( <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds <br/> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span>] ) ) )
 <b>by </b><i>, , , </i>;<br/>


</div>
