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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/> for <font color="Olive" title="b3">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b3">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> implies  for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/> ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/> ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:24"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font></span><br/>

<a NAME="E3:24"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E25">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> holds <br/> for <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">G</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c1">G</font></span><br/><a NAME="E2:24_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E3:24_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E4:24_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c1">G</font></span><br/><a NAME="E5:24_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font>;<br/><a NAME="E6:24_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font></span>)</span> <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="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="glib_001.html#D19" onmouseover="rs('glib_001/D19')" onmouseout="rh()">GLIB_001:def 19</a></span>;<br/><a NAME="E7:24_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">m</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:24_1"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T5" target="_self" onmouseover="rs('chord/T5')" onmouseout="rh()">Th5</a></span>;<br/><a NAME="E10:24_1"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="nat_1.html#T16" onmouseover="rs('nat_1/T16')" onmouseout="rh()">NAT_1:16</a></span>;<br/><a NAME="E11:24_1"/>
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="glib_001.html#D19" onmouseover="rs('glib_001/D19')" onmouseout="rh()">GLIB_001:def 19</a></span>;<br/><a NAME="E12:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="nat_1.html#T16" onmouseover="rs('nat_1/T16')" onmouseout="rh()">NAT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E13:24_1"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E14:24_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><a NAME="E15:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A15</font></span>: 
( <font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E16:24_1"/>
<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E13:24_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E17:24_1"/>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E10:24_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E15:24_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E5:24_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="glib_001.html#D19" onmouseover="rs('glib_001/D19')" onmouseout="rh()">GLIB_001:def 19</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="hsNdiv(this)" href="javascript:()" title="24_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2"/>
( <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:24_2_1"/>
<font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:24_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><span class="kw">hence </span><a NAME="E4:24_2_1"/>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:24_2_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="hsNdiv(this)" href="javascript:()" title="24_2_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_2"/><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:24_2_2"/>
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:24_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><a NAME="E4:24_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:24_2_2"><span class="lab"><font color="Green" title="E26">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:24_2_2"/>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">end;</span></div>

</div>
