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

<b>let </b><font color="Maroon" title="c1">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  <a href="abian.html#V2" title="ABIAN:attr.2">covering</a> <a href="taxonom2.html#V5" title="TAXONOM2:attr.5">T_3</a>  <a href="taxonom2.html#M1" title="TAXONOM2:mode.1">Hierarchy</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c2">H</font> is <a href="taxonom2.html#V6" title="TAXONOM2:attr.6">lower-bounded</a> &amp; not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be  <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:31"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green" title="E22">A4</font></i>: 
for <font color="Olive" title="b1">C</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">C</font> holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">C</font>
 ;<br/>

<a NAME="E5:31"/><i><font color="Green" title="E23">A5</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="abian.html#T4" title="ABIAN:th.4">ABIAN:4</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c2">H</font> is <a href="taxonom2.html#V3" title="TAXONOM2:attr.3">hierarchic</a>
 
<b>by </b><i><a class="ref" href="taxonom2.html#D4" target="_self" title="TAXONOM2:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1"/>
( <font color="Maroon" title="c1">Y</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>  or <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>suppose </b></a><a NAME="E1:31_1_1"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c1">Y</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> 
 ;<br/><div class="add"><a NAME="E2:31_1_1"/>
<font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1"/>
not <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1_1"/><i><font color="Green" title="E26">A8</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">Y</font>
 <b>and </b><br/><a NAME="E4:31_1_1_1"/><i><font color="Green" title="E27">A9</font></i>: 
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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> ] <b>means </b><font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c6">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:31_1_1_1"/><i><font color="Green" title="E28">A10</font></i>: 
for <font color="Olive" title="b1">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> iff ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">h</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c7">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:31_1_1_1"/><i><font color="Green" title="E29">A11</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">xx</font> &amp; <font color="Maroon" title="c7">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="taxonom2.html#E3:31_1_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E9:31_1_1_1"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c7">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:31_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="taxonom2.html#E8:31_1_1_1"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/>
<a NAME="E10:31_1_1_1"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">d</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="E1:31_1_1_1_1"/><i><font color="Green" title="E32">A14</font></i>: 
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 ;<br/>

<b>thus </b><a NAME="E2:31_1_1_1_1"/>
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:31_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_1"><i><font color="Green" title="E32">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">h1</font>, <font color="Maroon" title="c9">h2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1_1_2"/><i><font color="Green" title="E32">A15</font></i>: 
( <font color="Maroon" title="c8">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> &amp; <font color="Maroon" title="c9">h2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1_1_2_1"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c8">h1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">h2</font>
 ;<br/><a NAME="E2:31_1_1_1_2_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">h1</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">h2</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:31_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_2"><i><font color="Green" title="E32">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:31_1_1_1_2_1"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:31_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">h1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">h2</font> or <font color="Maroon" title="c9">h2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">h1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="taxonom2.html#E10:31_1_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_1_2"/>
<font color="Maroon" title="c8">h1</font>,<font color="Maroon" title="c9">h2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:31_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">D</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">min</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:31_1_1_1"/><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon" title="c8">min</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E15:31_1_1_1"/><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c8">min</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="taxonom2.html#E9:31_1_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="taxonom2.html#E10:31_1_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="taxonom2.html#D7" target="_self" title="TAXONOM2:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">min'</font> = <font color="Maroon" title="c8">min</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="taxonom2.html#E14:31_1_1_1"><i><font color="Green" title="E32">A17</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">b1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1_1_3"/><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon" title="c10">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">b1'</font> = <font color="Maroon" title="c10">b1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:31_1_1_1_3"><i><font color="Green" title="E34">A19</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="taxonom2.html#E3:31_1_1_1"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/><a NAME="E4:31_1_1_1_3"/><i><font color="Green" title="E35">A20</font></i>: 
not <font color="Maroon" title="c12">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">b1'</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:31_1_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_3"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E5:31_1_1_1_3"/><i><font color="Green" title="E36">A21</font></i>: 
not <font color="Maroon" title="c9">min'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1_3_1"/><i><font color="Green" title="E37">A22</font></i>: 
<font color="Maroon" title="c9">min'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">k</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:31_1_1_1_3_1"/><i><font color="Green" title="E38">A23</font></i>: 
<font color="Maroon" title="c12">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">k</font>
 <b>and </b><br/><a NAME="E4:31_1_1_1_3_1"/><i><font color="Green" title="E39">A24</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E5:31_1_1_1_3_1"/><i><font color="Green" title="E40">A25</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">b1'</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_3"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="taxonom2.html#E4:31_1_1_1_3"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="taxonom2.html#D6" target="_self" title="TAXONOM2:def.6">Def6</a></i>;<br/>
<a NAME="E6:31_1_1_1_3_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:31_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="taxonom2.html#E3:31_1_1_1_3_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="taxonom2.html#E4:31_1_1_1_3_1"><i><font color="Green" title="E39">A24</font></i></a></i>;<br/>
<a NAME="E7:31_1_1_1_3_1"/><b>then </b>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c6">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">k</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E8:31_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E41">A26</font></i>: 
<font color="Maroon" title="c9">min'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">k</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E15:31_1_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:31_1_1_1_3_1"/><i><font color="Green" title="E42">A27</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">min'</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="taxonom2.html#E14:31_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E11:31_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:31_1_1_1_3_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="taxonom2.html#E5:31_1_1_1_3_1"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="taxonom2.html#E8:31_1_1_1_3_1"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="taxonom2.html#E10:31_1_1_1_3_1"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:31_1_1_1_3"/>
not <font color="Maroon" title="c10">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">min'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1_3_2"/><i><font color="Green" title="E37">A28</font></i>: 
<font color="Maroon" title="c10">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">min'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">b1'</font> = <font color="Maroon" title="c10">b1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:31_1_1_1_3"><i><font color="Green" title="E34">A19</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">k</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E4:31_1_1_1_3_2"/><i><font color="Green" title="E38">A29</font></i>: 
<font color="Maroon" title="c12">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">k</font>
 <b>and </b><br/><a NAME="E5:31_1_1_1_3_2"/><i><font color="Green" title="E39">A30</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E6:31_1_1_1_3_2"/><i><font color="Green" title="E40">A31</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c13">b1'</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_3"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="taxonom2.html#E4:31_1_1_1_3"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="taxonom2.html#D6" target="_self" title="TAXONOM2:def.6">Def6</a></i>;<br/>
<a NAME="E7:31_1_1_1_3_2"/><i><font color="Green" title="E41">A32</font></i>: 
<font color="Maroon" title="c10">b1</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="taxonom2.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_3"><i><font color="Green" title="E34">A19</font></i></a></i>;<br/>
<a NAME="E8:31_1_1_1_3_2"/>
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:31_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="taxonom2.html#E4:31_1_1_1_3_2"><i><font color="Green" title="E38">A29</font></i></a>, <a class="txt" href="taxonom2.html#E5:31_1_1_1_3_2"><i><font color="Green" title="E39">A30</font></i></a></i>;<br/>
<a NAME="E9:31_1_1_1_3_2"/><b>then </b>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c6">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">k</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E10:31_1_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c9">min'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">k</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E15:31_1_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:31_1_1_1_3_2"/><b>then </b><i><font color="Green" title="E42">A33</font></i>: 
<font color="Maroon" title="c13">b1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">k</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:31_1_1_1_3_2"><i><font color="Green" title="E37">A28</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:31_1_1_1_3_2"/><i><font color="Green" title="E43">A34</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">b1'</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:31_1_1_1_3_2"><i><font color="Green" title="E41">A32</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E14:31_1_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:31_1_1_1_3_2"><i><font color="Green" title="E40">A31</font></i></a>, <a class="txt" href="taxonom2.html#E11:31_1_1_1_3_2"><i><font color="Green" title="E42">A33</font></i></a>, <a class="txt" href="taxonom2.html#E13:31_1_1_1_3_2"><i><font color="Green" title="E43">A34</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:31_1_1_1_3"/>
<font color="Maroon" title="c10">b1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">min'</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="taxonom2.html#E6:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="taxonom2.html#E14:31_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_3"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="taxonom2.html#E5:31_1_1_1_3"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:31_1_1_1"/><b>then </b><i><font color="Green" title="E34">A35</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/>( <font color="Maroon" title="c9">min'</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c1">Y</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="taxonom2.html#E1:31_1_1"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">C</font> = <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">min'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E19:31_1_1_1"/><i><font color="Green" title="E35">A36</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">min'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">min'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="taxonom2.html#E14:31_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="taxonom2.html#E18:31_1_1_1"><i><font color="Green" title="E34">A35</font></i></a>, <a class="ref" href="taxonom2.html#T12" target="_self" title="TAXONOM2:th.12">Th12</a></i>;<br/>
<a NAME="E20:31_1_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">min'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</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="E1:31_1_1_1_4"/><i><font color="Green" title="E36">A37</font></i>: 
<font color="Maroon" title="c11">s</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="c9">min'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:31_1_1_1_4"/>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:31_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_1_1_4"><i><font color="Green" title="E36">A37</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:31_1_1_1"/><b>then </b><i><font color="Green" title="E36">A38</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">min'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E22:31_1_1_1"/>
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">min'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E23:31_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="taxonom2.html#E4:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="taxonom2.html#E19:31_1_1_1"><i><font color="Green" title="E35">A36</font></i></a>, <a class="txt" href="taxonom2.html#E21:31_1_1_1"><i><font color="Green" title="E36">A38</font></i></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:31_1_1"/><b>then </b><i><font color="Green" title="E26">A39</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E4:31_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/>( <font color="Olive" title="b1">A</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>  &amp; ( for <font color="Olive" title="b2">E</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E</font> or <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">E</font> ) ) )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="taxonom2.html#D5" target="_self" title="TAXONOM2:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E5:31_1_1"/>
<font color="Maroon" title="c4">B</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:31_1_1"><i><font color="Green" title="E26">A39</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><b>suppose </b></a><a NAME="E1:31_1_2"/><i><font color="Green" title="E25">A40</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_2_1_1"/>
( <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">H</font> <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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:31_1_2"><i><font color="Green" title="E25">A40</font></i></a>, <a class="ref" href="zfmisc_1.html#T1" title="ZFMISC_1:th.1">ZFMISC_1:1</a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_2_1_1_1"/>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_2_1_1_1"/>
<font color="Maroon" title="c4">B</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="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_2_1_1_2"/><i><font color="Green" title="E26">A41</font></i>: 
<font color="Maroon" title="c2">H</font> <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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:31_1_2_1_1_2"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">B</font> <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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:31_1_2_1_1_2_1"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:31_1_2_1_1_2"/>
<font color="Maroon" title="c4">B</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="txt" href="taxonom2.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="taxonom2.html#E1:31_1_2_1_1_2"><i><font color="Green" title="E26">A41</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:31_1_2"/>
<font color="Maroon" title="c4">B</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:31_1_2"><i><font color="Green" title="E25">A40</font></i></a>, <a class="ref" href="eqrel_1.html#T54" title="EQREL_1:th.54">EQREL_1:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
