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

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E20">A2</font></i>: 
for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E4:23"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T26" target="_self" title="WELLORD2:th.26">Th26</a></i>;<br/>
<a NAME="E5:23"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E7:23"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E8:23"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$2,<font color="Olive" title="b2">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> ) );<br/>
<a NAME="E9:23"/><i><font color="Green" title="E26">A8</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:23_1"/><i><font color="Green" title="E28">A10</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; ( for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> ) )
 ;<br/>

<b>given </b><font color="Maroon" title="c7">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:23_1"/><i><font color="Green" title="E29">A11</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> &amp; ( for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Olive" title="b1">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> ) )
 ;<br/>

<a NAME="E4:23_1"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:23_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="wellord2.html#E2:23_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="wellord2.html#E3:23_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:23_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E5:23"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:23"/><i><font color="Green" title="E27">A12</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<a NAME="E2:23_2"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_2"/><i><font color="Green" title="E29">A14</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c4">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="txt" href="wellord2.html#E6:23"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c3">x</font>
;<br/>

<b>thus </b><a NAME="E5:23_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="txt" href="wellord2.html#E4:23_2"><i><font color="Green" title="E29">A14</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E6:23_2"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E7:23_2"/><b>then </b><i><font color="Green" title="E31">A16</font></i>: 
(  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
<b>by </b><i><a class="txt" href="wellord2.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="wellord2.html#E2:23_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="wellord2.html#E4:23_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E8:23_2"/><i><font color="Green" title="E32">A17</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
<b>by </b><i><a class="txt" href="wellord2.html#E8:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="wellord2.html#E2:23_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="wellord2.html#E6:23_2"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E9:23_2"/>
not <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:23_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="wellord2.html#E6:23_2"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E10:23_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E7:23_2"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="wellord2.html#E8:23_2"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:23"/><i><font color="Green" title="E28">A18</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="wellord2.html#E9:23"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="wellord2.html#E10:23"><i><font color="Green" title="E27">A12</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c4">Choice</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>;<br/>

<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E13:23"/><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font>;<br/>

<a NAME="E14:23"/><i><font color="Green" title="E30">A20</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_3"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<a NAME="E2:23_3"/><b>then </b>
 ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">X</font></span>)</span>,<font color="Olive" title="b2">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> ) )
 
<b>by </b><i><a class="txt" href="wellord2.html#E12:23"><i><font color="Green" title="E28">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:23_3"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E15:23"/>
<font color="Maroon" title="c4">Choice</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_4"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Choice</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">X</font>
 ;<br/>

<a NAME="E2:23_4"/><b>then </b><i><font color="Green" title="E31">A21</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Choice</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_4"/><i><font color="Green" title="E32">A22</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E5:23_4"/>
not <font color="Maroon" title="c7">y</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"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E6:23_4"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">z</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:23_4"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:23_4"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">z</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="wellord2.html#E12:23"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="wellord2.html#E13:23"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="wellord2.html#E4:23_4"><i><font color="Green" title="E32">A22</font></i></a></i>;<br/>
<a NAME="E8:23_4"/><b>then </b><i><font color="Green" title="E33">A23</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E9:23_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">z</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E12:23"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="wellord2.html#E14:23"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="wellord2.html#E4:23_4"><i><font color="Green" title="E32">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:23_4"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord2.html#E2:23_4"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="wellord2.html#E4:23_4"><i><font color="Green" title="E32">A22</font></i></a>, <a class="txt" href="wellord2.html#E8:23_4"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E16:23"/>
<font color="Maroon" title="c7">y</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"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E17:23"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:23"/><b>then </b>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Choice</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> )
 
<b>by </b><i><a class="txt" href="wellord2.html#E12:23"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="wellord2.html#E13:23"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="wellord2.html#E14:23"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E19:23"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Choice</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div>
