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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">G2</font>, <font color="Maroon" title="c4">H2</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H1</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) &amp; <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H2</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) holds <br/><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H1</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) &amp; <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H2</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) implies <font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c7">N</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>);<br/>
<a NAME="E1:151"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<a NAME="E2:151"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:151"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/>
<a NAME="E4:151"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E5:151"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E6:151"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:151"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E7:151"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E8:151"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:151"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:151"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E10:151"/><span class="lab"><font color="Green" title="E65">A10</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H1</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)
 <span class="kw">and </span><br/><a NAME="E11:151"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
<font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H2</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span><br/>

<a NAME="E12:151"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:151"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/>
<a NAME="E13:151"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E14:151"/><span class="lab"><font color="Green" title="E69">A14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:151"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/>
<a NAME="E15:151"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E16:151"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A15</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:151"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E5:151"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E7:151"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E13:151"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/>
<a NAME="E17:151"/><span class="lab"><font color="Green" title="E71">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:151"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:151_1"/><span class="lab"><font color="Green" title="E72">A17</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:151"><span class="lab"><font color="Green" title="E70">A15</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E73">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E74">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:151_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span><br/><a NAME="E2:151_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A20</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:151_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T135" target="_self" title="ZF_LANG1:th.135">Th148</a>, <a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/><a NAME="E4:151_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:151"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E1:151_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E2:151_1_1_1"><span class="lab"><font color="Green" title="E75">A20</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:151_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:151_1_1_1"><span class="lab"><font color="Green" title="E75">A20</font></span></a>, <a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:151_1_1"/><span class="lab"><font color="Green" title="E75">A21</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span><br/><a NAME="E3:151_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A22</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:151"><span class="lab"><font color="Green" title="E71">A16</font></span></a>, <a class="txt" href="#E8:151"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="txt" href="#E13:151"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:151_1_1_2"/><span class="lab"><font color="Green" title="E77">A23</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:151"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E1:151_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c10">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E3:151_1_1_2"/><span class="lab"><font color="Green" title="E78">A24</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G1</font>
 <span class="kw">and </span><br/><a NAME="E4:151_1_1_2"/><span class="lab"><font color="Green" title="E79">A25</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span><br/><a NAME="E5:151_1_1_2"/><span class="lab"><font color="Green" title="E80">A26</font></span>: 
( <font color="Maroon" title="c1">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c2">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:151"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E12:151"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:151_1_1"><span class="lab"><font color="Green" title="E76">A22</font></span></a>, <a class="txt" href="#E3:151_1_1_2"><span class="lab"><font color="Green" title="E78">A24</font></span></a>, <a class="txt" href="#E4:151_1_1_2"><span class="lab"><font color="Green" title="E79">A25</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E6:151_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A27</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:151"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E12:151"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:151_1_1_2"><span class="lab"><font color="Green" title="E78">A24</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/><a NAME="E7:151_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:151"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E12:151"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E3:151_1_1_2"><span class="lab"><font color="Green" title="E78">A24</font></span></a>, <a class="txt" href="#E5:151_1_1_2"><span class="lab"><font color="Green" title="E80">A26</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:151_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:151"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E1:151_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E6:151_1_1_2"><span class="lab"><font color="Green" title="E81">A27</font></span></a>, <a class="txt" href="#E1:151_1_1_2"><span class="lab"><font color="Green" title="E77">A23</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:151_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:151"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E1:151"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E2:151_1_1"><span class="lab"><font color="Green" title="E75">A21</font></span></a>, <a class="txt" href="#E1:151_1_1"><span class="lab"><font color="Green" title="E74">A19</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:151_1_2"/><span class="lab"><font color="Green" title="E74">A28</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:151"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E1:151_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c10">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E3:151_1_2"/><span class="lab"><font color="Green" title="E75">A29</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G2</font>
 <span class="kw">and </span><br/><a NAME="E4:151_1_2"/><span class="lab"><font color="Green" title="E76">A30</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span><br/><a NAME="E5:151_1_2"/><span class="lab"><font color="Green" title="E77">A31</font></span>: 
( <font color="Maroon" title="c3">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c4">H2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:151"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E6:151"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E14:151"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="txt" href="#E13:151"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E3:151_1_2"><span class="lab"><font color="Green" title="E75">A29</font></span></a>, <a class="txt" href="#E4:151_1_2"><span class="lab"><font color="Green" title="E76">A30</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E6:151_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A32</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:151"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E14:151"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="txt" href="#E3:151_1_2"><span class="lab"><font color="Green" title="E75">A29</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/><a NAME="E7:151_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:151"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E14:151"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="txt" href="#E3:151_1_2"><span class="lab"><font color="Green" title="E75">A29</font></span></a>, <a class="txt" href="#E5:151_1_2"><span class="lab"><font color="Green" title="E77">A31</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:151_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:151"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E1:151_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E1:151_1_2"><span class="lab"><font color="Green" title="E74">A28</font></span></a>, <a class="txt" href="#E6:151_1_2"><span class="lab"><font color="Green" title="E78">A32</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:151_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:151"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E16:151"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="txt" href="#E1:151_1"><span class="lab"><font color="Green" title="E72">A17</font></span></a>, <a class="txt" href="#E3:151_1"><span class="lab"><font color="Green" title="E73">A18</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E19:151"/>
<font color="Maroon" title="c1">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:151"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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