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

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">sq</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<a NAME="E1:44"/><i><font color="Green">E31</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/>for <font color="Olive">H</font>, <font color="Olive">F</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">sq</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">sq</font> holds <br/><font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> ) holds <br/>for <font color="Olive">H</font>, <font color="Olive">F</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">sq</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">sq</font> holds <br/><font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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="E1:44_1"/><i><font color="Green">E32</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">H</font>, <font color="Olive">F</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">sq</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">sq</font> holds <br/><font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font>
 ;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">sq</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E2:44_1"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font> )
 ;<br/>

<a NAME="E3:44_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E4:44_1"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:44_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:44_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:44_1"/><b>then </b><i><font color="Green">E41</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="zf_lang.html#E2:44_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V7" target="_self">atomic</a>
 ;<br/><a NAME="E2:44_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i/>;<br/><a NAME="E3:44_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="ref" href="zf_lang.html#T12" target="_self">Th12</a>, , <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:44_1_1"/><b>then </b>
3 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font></span>)</span> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="ref" href="zf_lang.html#T12" target="_self">Th12</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:44_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E6:44_1_1"/><b>then </b>
<font color="Maroon">sq</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E7:44_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V4" target="_self">negative</a>
 ;<br/><b>then consider </b><font color="Maroon">H1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:44_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/><a NAME="E4:44_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, , </i>;<br/><a NAME="E5:44_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="txt" href="zf_lang.html#E3:44_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:44_1_2"/><b>then </b>
<font color="Maroon">F</font> is <a href="zf_lang.html#V4" target="_self">negative</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E8:44_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">F1</font>
 <b>by </b><i/>;<br/><a NAME="E9:44_1_2"/>
(  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <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#K8">^</a> <font color="Maroon">H1</font> &amp;  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <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#K8">^</a> <font color="Maroon">F1</font> &amp; <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#K8">^</a> <font color="Maroon">F1</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font> <a href="hidden.html#R1">=</a> <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">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E10:44_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E4:44_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E11:44_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <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></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <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="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="zf_lang.html#E4:44_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E12:44_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E13:44_1_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E4:44_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_lang.html#E8:44_1"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V6" target="_self">universal</a>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">H1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:44_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">H1</font>
 <b>by </b><i/>;<br/><a NAME="E4:44_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 4
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E4:44_1"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E5:44_1_3"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 4
 <b>by </b><i><a class="txt" href="zf_lang.html#E3:44_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:44_1_3"/><b>then </b>
<font color="Maroon">F</font> is <a href="zf_lang.html#V6" target="_self">universal</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E8:44_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8" target="_self">All</a> <font color="Maroon">y</font>,<font color="Maroon">F1</font>
 <b>by </b><i/>;<br/><a NAME="E9:44_1_3"/><i><font color="Green">E89</font></i>: 
(  <a href="zf_lang.html#K8" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">H1</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">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font> &amp;  <a href="zf_lang.html#K8" target="_self">All</a> <font color="Maroon">y</font>,<font color="Maroon">F1</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">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F1</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</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">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</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"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E10:44_1_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_lang.html#E8:44_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E11:44_1_3"/>
( <span class="p1">(<span class="default"><span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E12:44_1_3"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E9:44_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E13:44_1_3"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">4,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">4,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/><a NAME="E14:44_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E15:44_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E16:44_1_3"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E8:44_1"><i><font color="Green">E42</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a>
 ;<br/><b>then consider </b><font color="Maroon">G1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">G</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:44_1_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E4:44_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 3 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E4:44_1"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E5:44_1_4"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="txt" href="zf_lang.html#E3:44_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:44_1_4"/><b>then </b>
<font color="Maroon">F</font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">H1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E8:44_1_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/><a NAME="E9:44_1_4"/>
( <font color="Maroon">G1</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Maroon">G</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">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Maroon">H1</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">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</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">F1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E10:44_1_4"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">G1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_lang.html#E8:44_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E11:44_1_4"/><b>then </b><i><font color="Green">E96</font></i>: 
( (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> implies  ex <font color="Olive">sq'</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">sq'</font> ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> implies  ex <font color="Olive">sq'</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">sq'</font> ) )
 <b>by </b><i><a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_4_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">sq'</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><a NAME="E1:44_1_4_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq'</font>
 ;<br/><a NAME="E2:44_1_4_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E3:44_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:44_1_4_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E5:44_1_4_1"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_4_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">sq'</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><a NAME="E1:44_1_4_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq'</font>
 ;<br/><a NAME="E2:44_1_4_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_lang.html#E8:44_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E3:44_1_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:44_1_4_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E5:44_1_4_2"/>
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, </i>;<br/></div><b>end;</b></div><a NAME="E14:44_1_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font>
 <b>by </b><i>, <a class="txt" href="zf_lang.html#E9:44_1"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E15:44_1_4"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E16:44_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E17:44_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E18:44_1_4"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_lang.html#T10" target="_self">Th10</a>, <a class="txt" href="zf_lang.html#E7:44_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_lang.html#E8:44_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="zf_lang.html#E9:44_1"><i><font color="Green">E44</font></i></a>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:44_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self">Def8</a>, , , </i>;<br/>


</div><b>end;</b></div>
<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">H</font>, <font color="Olive">F</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">sq</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">sq</font> holds <br/><font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font>;<br/>
<a NAME="E2:44"/><i><font color="Green">E102</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 ( 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="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i><a class="txt" href="zf_lang.html#E1:44"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:44"/><i><font color="Green">E103</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="E4:44"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
;<br/>
<b>hence </b><a NAME="E5:44"/>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sq</font> implies <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T10" target="_self">Th10</a></i>;<br/>


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