<?xml version="1.0"?>
<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>let </b><font color="Maroon">t</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>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> <br/> for <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">t</font> holds <br/><font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font>;<br/>
<a NAME="E1:26"/><i><font color="Green">E29</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>   st ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_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="E1:26_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/>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> <br/> for <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">t</font> holds <br/><font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font>
 ;<br/>

<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>let </b><font color="Maroon">t</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E2:26_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:26_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E4:26_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E5:26_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon">p</font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a> or <font color="Maroon">p</font> is <a href="hilbert2.html#V2" target="_self">conditional</a> or <font color="Maroon">p</font> is <a href="hilbert2.html#V3" target="_self">simple</a> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:26_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</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="E4:26_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>then </b><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/><a NAME="E6:26_1_1_1"/><b>then </b>
<font color="Maroon">q</font> is <a href="hilbert2.html#V1" target="_self">conjunctive</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E8:26_1_1_1"/><i><font color="Green">E39</font></i>: 
<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>
 <b>by </b><i/>;<br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s'</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, , <a class="ref" href="hilbert2.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><a NAME="E10:26_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E11:26_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">s'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_3_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:26_1_1_1_3_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:26_1_1_1_3_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:26_1_1_1_3_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:26_1_1_1_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:26_1_1_1_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:26_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:26_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">s'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E10:26_1_1_1_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:26_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="hilbert2.html#E1:26_1"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E12:26_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:26_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="trees_1.html#T5">TREES_1:5</a></i>;<br/><b>hence </b><a NAME="E13:26_1_1_1_3_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:26_1_1_1_3_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:26_1_1_1_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:26_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:26_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">s'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E7:26_1_1_1_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:26_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="hilbert2.html#E1:26"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E9:26_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="trees_1.html#T5">TREES_1:5</a></i>;<br/><b>hence </b><a NAME="E10:26_1_1_1_3_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:26_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V2" target="_self">conditional</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E3:26_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</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="E4:26_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>then </b><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/><a NAME="E6:26_1_1_2"/><b>then </b>
<font color="Maroon">q</font> is <a href="hilbert2.html#V2" target="_self">conditional</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E8:26_1_1_2"/><i><font color="Green">E90</font></i>: 
<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>
 <b>by </b><i/>;<br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s'</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, , <a class="ref" href="hilbert2.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><a NAME="E10:26_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E11:26_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">s'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_2_3_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_2_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:26_1_1_2_3_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:26_1_1_2_3_1_1"/><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:26_1_1_2_3_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:26_1_1_2_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:26_1_1_2_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:26_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:26_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">s'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E10:26_1_1_2_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:26_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="hilbert2.html#E1:26_1"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E12:26_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:26_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="trees_1.html#T5">TREES_1:5</a></i>;<br/><b>hence </b><a NAME="E13:26_1_1_2_3_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:26_1_1_2_3_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">r'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:26_1_1_2_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:26_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:26_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">s'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E7:26_1_1_2_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:26_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="hilbert2.html#E1:26"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E9:26_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="hilbert2.html#E1:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="trees_1.html#T5">TREES_1:5</a></i>;<br/><b>hence </b><a NAME="E10:26_1_1_2_3_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:26_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_3"><b>suppose </b></a><a NAME="E1:26_1_1_3"/>
<font color="Maroon">p</font> is <a href="hilbert2.html#V3" target="_self">simple</a>
 ;<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="E3:26_1_1_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E4:26_1_1_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:26_1_1_3"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E6:26_1_1_3"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E7:26_1_1_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="hilbert2.html#T1" target="_self">Th1</a>, <a class="ref" href="trees_1.html#T6">TREES_1:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_4"><b>suppose </b></a><a NAME="E1:26_1_1_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="hilbert1.html#K2">VERUM</a> 
 ;<br/><div class="add"><a NAME="E2:26_1_1_4"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:26_1_1_4"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:26_1_1_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="trees_1.html#T6">TREES_1:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:26"/><i><font color="Green">E100</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>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<a NAME="E3:26"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
;<br/>
<b>hence </b><a NAME="E4:26"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> implies <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>


</div>
