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

<b>let </b><font color="Maroon">G1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H2</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>











<b>set </b><font color="Maroon">N</font> = <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>;<br/>
<a NAME="E1:154"/><i><font color="Green">E46</font></i>: 
(  <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#K4">dom</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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E2:154"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</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">G1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G2</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">G2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H1</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">H1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H2</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">H2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:154"/><i><font color="Green">E51</font></i>: 
(  <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> 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> &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">H1</font></span>)</span> <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> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T69" target="_self">Th69</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E4:154"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></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">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">G2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E5:154"/><i><font color="Green">E65</font></i>: 
(  <a href="finseq_1.html#K4">dom</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>  <a href="finseq_1.html#K2">Seg</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;  <a href="finseq_1.html#K4">dom</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">H1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</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> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T72" target="_self">Th72</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E6:154"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H2</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E7:154"/><b>then </b><i><font color="Green">E67</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H2</font> )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E4:154"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E8:154"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:154"/><b>then </b><i><font color="Green">E69</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">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T71" target="_self">Th71</a>, <a class="ref" href="zf_lang1.html#T74" target="_self">Th74</a>, <a class="txt" href="zf_lang1.html#E4:154"><i><font color="Green">E52</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:154_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="zf_lang1.html#E7:154"><i><font color="Green">E67</font></i></a></i>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:154_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>
 ;<br/><a NAME="E2:154_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">H1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="txt" href="zf_lang1.html#E6:154"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:154_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:154_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T33">ZF_LANG:33</a></i>;<br/><a NAME="E3:154_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="txt" href="zf_lang1.html#E7:154"><i><font color="Green">E67</font></i></a>, <a class="txt" href="zf_lang1.html#E8:154"><i><font color="Green">E68</font></i></a>, <a class="txt" href="zf_lang1.html#E4:154"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zf_lang.html#T33">ZF_LANG:33</a></i>;<br/><b>hence </b><a NAME="E4:154_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:154_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E2:154_1_1_2"/>
( <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">H1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T69" target="_self">Th69</a>, <a class="txt" href="zf_lang1.html#E5:154"><i><font color="Green">E65</font></i></a>, <a class="txt" href="zf_lang1.html#E3:154_1"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:154_1_1_2"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 <b>by </b><i>, <a class="txt" href="zf_lang1.html#E7:154"><i><font color="Green">E67</font></i></a>, <a class="txt" href="zf_lang1.html#E8:154"><i><font color="Green">E68</font></i></a>, <a class="txt" href="zf_lang1.html#E4:154"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E4:154_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:154_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T69" target="_self">Th69</a>, <a class="txt" href="zf_lang1.html#E1:154_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="zf_lang1.html#E1:154_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:154_1_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">i</font> <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">G1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E2:154_1_2"/>
( <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">H2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T71" target="_self">Th71</a>, <a class="ref" href="zf_lang1.html#T72" target="_self">Th72</a>, <a class="txt" href="zf_lang1.html#E6:154"><i><font color="Green">E66</font></i></a>, <a class="txt" href="zf_lang1.html#E1:154_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:154_1_2"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> or <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 <b>by </b><i>, <a class="txt" href="zf_lang1.html#E7:154"><i><font color="Green">E67</font></i></a>, <a class="txt" href="zf_lang1.html#E8:154"><i><font color="Green">E68</font></i></a>, <a class="txt" href="zf_lang1.html#E4:154"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E4:154_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:154_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T72" target="_self">Th72</a>, <a class="txt" href="zf_lang1.html#E7:154"><i><font color="Green">E67</font></i></a>, <a class="txt" href="zf_lang1.html#E8:154"><i><font color="Green">E68</font></i></a>, <a class="txt" href="zf_lang1.html#E9:154"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:154"/>
<font color="Maroon">G1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E7:154"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
