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

<b>let </b><font color="Maroon" title="c1">X</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>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">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">H</font> is <a href="taxonom2.html#V6" title="TAXONOM2:attr.6">lower-bounded</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E25">A2</font></i>: 
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>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green" title="E26">A3</font></i>: 
for <font color="Olive" title="b1">C1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">C1</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; <font color="Olive" title="b1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">P2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; <font color="Olive" title="b3">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; not <font color="Olive" title="b2">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b3">P2</font> holds <br/><font color="Olive" title="b3">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">P1</font> ) holds <br/> ex <font color="Olive" title="b2">PX</font>, <font color="Olive" title="b3">PY</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">PX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; <font color="Olive" title="b3">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; ( for <font color="Olive" title="b4">PZ</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> holds <br/>( <font color="Olive" title="b4">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b3">PY</font> &amp; <font color="Olive" title="b2">PX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b4">PZ</font> ) ) )
 ;<br/>

<a NAME="E4:37"/><i><font color="Green" title="E27">A4</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/>
<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>( ( for <font color="Olive" title="b1">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/><font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font> ) &amp; ( for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b2">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; not <font color="Olive" title="b1">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">P2</font> holds <br/><font color="Olive" title="b2">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">P1</font> ) );<br/>
<b>consider </b><font color="Maroon" title="c3">RL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green" title="E28">A5</font></i>: 
for <font color="Olive" title="b1">L</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RL</font> iff ( <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">L</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>
<a NAME="E7:37"/>
 <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">X</font>
 
<b>by </b><i><a class="ref" href="abian.html#T4" title="ABIAN:th.4">ABIAN:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:37"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c4">h</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="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">h</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">PX</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E12:37"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">PX</font>
 <b>and </b><br/><a NAME="E13:37"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c6">PX</font> <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#E1:37"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="taxonom2.html#E2:37"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="taxonom2.html#E9:37"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="taxonom2.html#T17" target="_self" title="TAXONOM2:th.17">Th17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">L</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E14:37"/><i><font color="Green" title="E31">A8</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">l</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="E1:37_1"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c8">l</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">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:37_1"/>
<font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">PX</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:37"/><i><font color="Green" title="E32">A10</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">a</font> <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#E13:37"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">P1</font>, <font color="Maroon" title="c9">P2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2"/>
( <font color="Maroon" title="c8">P1</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">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c9">P2</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">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><a NAME="E2:37_2"/><b>then </b>
( <font color="Maroon" title="c8">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">PX</font> &amp; <font color="Maroon" title="c9">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">PX</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="E3:37_2"/>
( <font color="Maroon" title="c8">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c9">P2</font> or <font color="Maroon" title="c9">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c8">P1</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E17:37"/><b>then </b><i><font color="Green" title="E33">A11</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RL</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E14:37"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green" title="E32">A10</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:37_3"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c8">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">RL</font>
 <b>and </b><br/><a NAME="E2:37_3"/><i><font color="Green" title="E35">A13</font></i>: 
<font color="Maroon" title="c8">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 ;<br/><div><i><font color="Green" title="E36">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">X1</font>, <font color="Maroon" title="c10">X2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_1"/>
( <font color="Maroon" title="c9">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c10">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> )
 ;<br/><a NAME="E2:37_3_1"/><b>then </b>
<font color="Maroon" title="c9">X1</font>,<font color="Maroon" title="c10">X2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_3"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:37_3_1"/>
( <font color="Maroon" title="c9">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">X2</font> or <font color="Maroon" title="c10">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">X1</font> )
 <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><b>set </b><font color="Maroon" title="c9">Y</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</font>;<br/><b>take </b><font color="Maroon" title="c10">Y</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</font>;<br/><a NAME="E4:37_3"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">P</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="E1:37_3_2"/><i><font color="Green" title="E38">A16</font></i>: 
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">L4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_3_2"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">L4</font> &amp; <font color="Maroon" title="c12">L4</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="taxonom2.html#E1:37_3_2"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:37_3_2"/>
<font color="Maroon" title="c12">L4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_2"><i><font color="Green" title="E39">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37_3_2"/>
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_3_2"><i><font color="Green" title="E39">A17</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E38">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_3"/>
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">L3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_3_3"/><i><font color="Green" title="E39">A19</font></i>: 
( <font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">L3</font> &amp; <font color="Maroon" title="c12">L3</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="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>thus </b><a NAME="E4:37_3_3"/>
<font color="Maroon" title="c11">P</font> <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#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_3"><i><font color="Green" title="E39">A19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">P1</font>, <font color="Maroon" title="c12">P2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_4"/><i><font color="Green" title="E39">A20</font></i>: 
( <font color="Maroon" title="c11">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c12">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c13">L1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_3_4"/><i><font color="Green" title="E40">A21</font></i>: 
( <font color="Maroon" title="c11">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">L1</font> &amp; <font color="Maroon" title="c13">L1</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="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">L2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:37_3_4"/><i><font color="Green" title="E41">A22</font></i>: 
( <font color="Maroon" title="c12">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">L2</font> &amp; <font color="Maroon" title="c14">L2</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="taxonom2.html#E1:37_3_4"><i><font color="Green" title="E39">A20</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_3_4_1"/>
( <font color="Maroon" title="c13">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">L2</font> or <font color="Maroon" title="c14">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L1</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E3:37_3"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_4"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3_4"><i><font color="Green" title="E41">A22</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_4_1_1"><b>suppose </b></a><a NAME="E1:37_3_4_1_1"/>
<font color="Maroon" title="c13">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">L2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_3_4_1_1"/>
( <font color="Maroon" title="c11">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c12">P2</font> or <font color="Maroon" title="c12">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_4"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3_4"><i><font color="Green" title="E41">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_4_1_2"><b>suppose </b></a><a NAME="E1:37_3_4_1_2"/>
<font color="Maroon" title="c14">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_3_4_1_2"/>
( <font color="Maroon" title="c11">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c12">P2</font> or <font color="Maroon" title="c12">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_4"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3_4"><i><font color="Green" title="E41">A22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:37_3"/>
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RL</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_3"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3"><i><font color="Green" title="E38">A18</font></i></a></i>;<br/><b>thus </b><a NAME="E8:37_3"/>
for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:37"/><i><font color="Green" title="E34">A23</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RL</font>
 <b>and </b><br/><a NAME="E21:37"/><i><font color="Green" title="E35">A24</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RL</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">C</font> holds <br/> not <font color="Maroon" title="c8">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E17:37"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="orders_1.html#T175" title="ORDERS_1:th.175">ORDERS_1:175</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">C</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">C</font>
;<br/>

<a NAME="E23:37"/><i><font color="Green" title="E36">A25</font></i>: 
<font color="Maroon" title="c9">C</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">P1</font>, <font color="Maroon" title="c11">P2</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_4"/><i><font color="Green" title="E37">A26</font></i>: 
<font color="Maroon" title="c10">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>and </b><br/><a NAME="E2:37_4"/><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c11">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/>

<b>thus </b><a NAME="E3:37_4"/>
( <font color="Maroon" title="c10">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P2</font> or <font color="Maroon" title="c11">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c10">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_4"><i><font color="Green" title="E37">A26</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_4"><i><font color="Green" title="E38">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:37"/><i><font color="Green" title="E37">A28</font></i>: 
<font color="Maroon" title="c9">C</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#E17:37"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="taxonom2.html#E21:37"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E25:37"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_5"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
  <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="37_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">h</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="E1:37_5_1"/><i><font color="Green" title="E38">A29</font></i>: 
<font color="Maroon" title="c10">h</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="c9">C</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_1"/><i><font color="Green" title="E39">A30</font></i>: 
( <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font> &amp; <font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_1"><i><font color="Green" title="E38">A29</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:37_5_1"/>
<font color="Maroon" title="c11">P</font> <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#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_1"><i><font color="Green" title="E39">A30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37_5_1"/>
<font color="Maroon" title="c10">h</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#E3:37_5_1"><i><font color="Green" title="E39">A30</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:37_5"/>
<font color="Maroon" title="c2">H</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="c9">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">h</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="E1:37_5_2"/><i><font color="Green" title="E38">A31</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1"/>
(  not <font color="Maroon" title="c10">h</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="c9">C</font> or <font color="Maroon" title="c10">h</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="c9">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1"/><i><font color="Green" title="E39">A32</font></i>: 
not <font color="Maroon" title="c10">h</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="c9">C</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c11">PS</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:37_5_2_1_1"/><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">PS</font>
 <b>and </b><br/><a NAME="E4:37_5_2_1_1"/><i><font color="Green" title="E41">A34</font></i>: 
<font color="Maroon" title="c11">PS</font> <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#E1:37"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="taxonom2.html#E2:37"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="taxonom2.html#T17" target="_self" title="TAXONOM2:th.17">Th17</a></i>;<br/><a NAME="E5:37_5_2_1_1"/><i><font color="Green" title="E42">A35</font></i>: 
<font color="Maroon" title="c10">h</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#E3:37_5_2_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">hx</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">hx</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">hx</font> );<br/><b>consider </b><font color="Maroon" title="c12">Ca</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:37_5_2_1_1"/><i><font color="Green" title="E43">A36</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</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><a NAME="E8:37_5_2_1_1"/><i><font color="Green" title="E44">A37</font></i>: 
<font color="Maroon" title="c12">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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="E1:37_5_2_1_1_1"/><i><font color="Green" title="E45">A38</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font>
 ;<br/>

<b>thus </b><a NAME="E2:37_5_2_1_1_1"/>
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_5_2_1_1"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_1"><i><font color="Green" title="E45">A38</font></i></a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">hx</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b1">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">hx</font> );<br/><b>consider </b><font color="Maroon" title="c13">Cb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:37_5_2_1_1"/><i><font color="Green" title="E45">A39</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p</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><a NAME="E11:37_5_2_1_1"/><i><font color="Green" title="E46">A40</font></i>: 
<font color="Maroon" title="c13">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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="E1:37_5_2_1_1_2"/><i><font color="Green" title="E47">A41</font></i>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font>
 ;<br/>

<b>thus </b><a NAME="E2:37_5_2_1_1_2"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_5_2_1_1"><i><font color="Green" title="E45">A39</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_2"><i><font color="Green" title="E47">A41</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E47">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_3"/><i><font color="Green" title="E48">A43</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><b>assume </b><a NAME="E2:37_5_2_1_1_3"/><i><font color="Green" title="E49">A44</font></i>: 
for <font color="Olive" title="b1">hv</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">hv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">p</font> holds <br/><font color="Olive" title="b1">hv</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font>
 ;<br/><b>consider </b><font color="Maroon" title="c15">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:37_5_2_1_1_3"/><i><font color="Green" title="E50">A45</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:37_5_2_1_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:37_5_2_1_1_3"/>
<font color="Maroon" title="c14">p</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_3"><i><font color="Green" title="E48">A43</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><a NAME="E6:37_5_2_1_1_3"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:37_5_2_1_1_3"/><i><font color="Green" title="E51">A46</font></i>: 
( <font color="Maroon" title="c15">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">v</font> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">p</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_5_2_1_1_3"><i><font color="Green" title="E50">A45</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E9:37_5_2_1_1_3"/>
not <font color="Maroon" title="c16">v</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37_5_2_1_1_3"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_5_2_1_1_3"><i><font color="Green" title="E51">A46</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="E10:37_5_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_5_2_1_1_3"><i><font color="Green" title="E49">A44</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_5_2_1_1_3"><i><font color="Green" title="E51">A46</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E13:37_5_2_1_1"/><i><font color="Green" title="E48">A47</font></i>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">Cb</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_5_2_1_1_4"/>
<font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">Cb</font>
  <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="37_5_2_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">p</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="E1:37_5_2_1_1_4_1"/><i><font color="Green" title="E49">A48</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">hv</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_2_1_1_4_1"/><i><font color="Green" title="E50">A49</font></i>: 
( <font color="Maroon" title="c15">hv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">p</font> &amp; not <font color="Maroon" title="c15">hv</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E12:37_5_2_1_1"><i><font color="Green" title="E47">A42</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_4_1"><i><font color="Green" title="E49">A48</font></i></a></i>;<br/>
<a NAME="E4:37_5_2_1_1_4_1"/><i><font color="Green" title="E51">A50</font></i>: 
<font color="Maroon" title="c14">p</font> <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#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_4_1"><i><font color="Green" title="E49">A48</font></i></a></i>;<br/>
<a NAME="E5:37_5_2_1_1_4_1"/><i><font color="Green" title="E52">A51</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">hv</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_4_1"><i><font color="Green" title="E49">A48</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_4_1"><i><font color="Green" title="E50">A49</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_4_1_1"/>
( <font color="Maroon" title="c15">hv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> or <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">hv</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_4_1"><i><font color="Green" title="E50">A49</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_5_2_1_1_4_1"><i><font color="Green" title="E51">A50</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_4_1_1_1"/>
<font color="Maroon" title="c15">hv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_5_2_1_1"><i><font color="Green" title="E45">A39</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_4_1"><i><font color="Green" title="E49">A48</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_4_1"><i><font color="Green" title="E50">A49</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1_4_1"><i><font color="Green" title="E52">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_5_2_1_1_4_1_1_1"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">Cb</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_4_1_1_2"/>
<font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">hv</font>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_5_2_1_1"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_4_1"><i><font color="Green" title="E49">A48</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_4_1"><i><font color="Green" title="E50">A49</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1_4_1"><i><font color="Green" title="E52">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_5_2_1_1_4_1_1_2"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">Cb</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:37_5_2_1_1_4"/>
<font color="Maroon" title="c12">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E8:37_5_2_1_1"><i><font color="Green" title="E44">A37</font></i></a>, <a class="txt" href="taxonom2.html#E11:37_5_2_1_1"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:37_5_2_1_1"/><b>then </b><i><font color="Green" title="E49">A52</font></i>: 
<font color="Maroon" title="c12">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E15:37_5_2_1_1"/><i><font color="Green" title="E50">A53</font></i>: 
<font color="Maroon" title="c13">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_5_2_1_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E16:37_5_2_1_1"/><i><font color="Green" title="E51">A54</font></i>: 
<font color="Maroon" title="c12">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_5_2_1_1"><i><font color="Green" title="E49">A52</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green" title="E52">A55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">P1</font>, <font color="Maroon" title="c15">P2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_5"/><i><font color="Green" title="E53">A56</font></i>: 
( <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> )
 ;<br/><a NAME="E2:37_5_2_1_1_5"/>
( <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_5_2_1_1"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_5"><i><font color="Green" title="E53">A56</font></i></a></i>;<br/><a NAME="E3:37_5_2_1_1_5"/><b>then </b>
( <font color="Maroon" title="c14">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c15">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_5_2_1_1_5"/>
( <font color="Maroon" title="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> or <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="taxonom2.html#E14:37_5_2_1_1"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_5"><i><font color="Green" title="E53">A56</font></i></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E18:37_5_2_1_1"/><i><font color="Green" title="E53">A57</font></i>: 
<font color="Maroon" title="c13">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_5_2_1_1"><i><font color="Green" title="E50">A53</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green" title="E54">A58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">P1</font>, <font color="Maroon" title="c15">P2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_6"/><i><font color="Green" title="E55">A59</font></i>: 
( <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font> )
 ;<br/><a NAME="E2:37_5_2_1_1_6"/>
( <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_5_2_1_1"><i><font color="Green" title="E50">A53</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_6"><i><font color="Green" title="E55">A59</font></i></a></i>;<br/><a NAME="E3:37_5_2_1_1_6"/><b>then </b>
( <font color="Maroon" title="c14">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c15">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_5_2_1_1_6"/>
( <font color="Maroon" title="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> or <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="taxonom2.html#E15:37_5_2_1_1"><i><font color="Green" title="E50">A53</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_6"><i><font color="Green" title="E55">A59</font></i></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1"/>
( <font color="Maroon" title="c13">Cb</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="c13">Cb</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="37_5_2_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1"/>
<font color="Maroon" title="c13">Cb</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"><b>then consider </b><font color="Maroon" title="c14">PX</font>, <font color="Maroon" title="c15">Pmax</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E55">A60</font></i>: 
( <font color="Maroon" title="c14">PX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font> &amp; <font color="Maroon" title="c15">Pmax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font> )
 <b>and </b><br/><a NAME="E4:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E56">A61</font></i>: 
for <font color="Olive" title="b1">PZ</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font> holds <br/>( <font color="Olive" title="b1">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Maroon" title="c14">PX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">PZ</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="taxonom2.html#E18:37_5_2_1_1"><i><font color="Green" title="E53">A57</font></i></a>, <a class="txt" href="taxonom2.html#E19:37_5_2_1_1"><i><font color="Green" title="E54">A58</font></i></a></i>;<br/><a NAME="E5:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E57">A62</font></i>: 
<font color="Maroon" title="c15">Pmax</font> <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#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="taxonom2.html#E15:37_5_2_1_1"><i><font color="Green" title="E50">A53</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1"><i><font color="Green" title="E55">A60</font></i></a></i>;<br/><a NAME="E6:37_5_2_1_1_7_1_1"/>
<font color="Maroon" title="c15">Pmax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_5_2_1_1"><i><font color="Green" title="E50">A53</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1"><i><font color="Green" title="E55">A60</font></i></a></i>;<br/><a NAME="E7:37_5_2_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E58">A63</font></i>: 
<font color="Maroon" title="c15">Pmax</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font>;<br/><b>consider </b><font color="Maroon" title="c16">Pb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E59">A64</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Pb</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</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>set </b><font color="Maroon" title="c17">PS1</font> = <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E10:37_5_2_1_1_7_1_1"/>
<font color="Maroon" title="c16">Pb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Pmax</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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="E1:37_5_2_1_1_7_1_1_1"/>
<font color="Maroon" title="c18">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Pb</font>
 ;<br/>

<b>hence </b><a NAME="E2:37_5_2_1_1_7_1_1_1"/>
<font color="Maroon" title="c18">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_1"><i><font color="Green" title="E59">A64</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:37_5_2_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E60">A65</font></i>: 
<font color="Maroon" title="c16">Pb</font> <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#E5:37_5_2_1_1_7_1_1"><i><font color="Green" title="E57">A62</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E61">A66</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> or <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> or <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> or <font color="Maroon" title="c10">h</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1_7_1_1"><i><font color="Green" title="E57">A62</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></i>;<br/><a NAME="E13:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E62">A67</font></i>: 
 ex <font color="Olive" title="b1">hx</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Olive" title="b1">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">hx</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_5_2_1_1"><i><font color="Green" title="E45">A39</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1"><i><font color="Green" title="E55">A60</font></i></a></i>;<br/><a NAME="E14:37_5_2_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E63">A68</font></i>: 
<font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_5_2_1_1_7_1_1"><i><font color="Green" title="E58">A63</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_1"><i><font color="Green" title="E59">A64</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_5_2_1_1_7_1_1"><i><font color="Green" title="E61">A66</font></i></a>, <a class="ref" href="taxonom2.html#T19" target="_self" title="TAXONOM2:th.19">Th19</a></i>;<br/><a NAME="E15:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E64">A69</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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="E1:37_5_2_1_1_7_1_1_2"/><i><font color="Green" title="E65">A70</font></i>: 
<font color="Maroon" title="c18">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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:37_5_2_1_1_7_1_1_2"/>
<font color="Maroon" title="c18">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_2"><i><font color="Green" title="E65">A70</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_5_2_1_1_7_1_1_2"/>
<font color="Maroon" title="c18">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_5_2_1_1_7_1_1"><i><font color="Green" title="E63">A68</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E65">A71</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</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="37_5_2_1_1_7_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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="E1:37_5_2_1_1_7_1_1_3"/><i><font color="Green" title="E66">A72</font></i>: 
<font color="Maroon" title="c18">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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:37_5_2_1_1_7_1_1_3"/>
<font color="Maroon" title="c18">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#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_3"><i><font color="Green" title="E66">A72</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="E17:37_5_2_1_1_7_1_1"/>
<font color="Maroon" title="c10">h</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E18:37_5_2_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E66">A73</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E19:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E67">A74</font></i>: 
<font color="Maroon" title="c16">Pb</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="c10">h</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#E11:37_5_2_1_1_7_1_1"><i><font color="Green" title="E60">A65</font></i></a>, <a class="txt" href="taxonom2.html#E16:37_5_2_1_1_7_1_1"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>set </b><font color="Maroon" title="c18">C1</font> = <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E20:37_5_2_1_1_7_1_1"/>
<font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E21:37_5_2_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E68">A75</font></i>: 
<font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E22:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E69">A76</font></i>: 
<font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_5_2_1_1_7_1_1"><i><font color="Green" title="E64">A69</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><div><i><font color="Green" title="E70">A77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">P3</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_7_1_1_4"/><i><font color="Green" title="E71">A78</font></i>: 
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_1_4_1"/>
( <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c19">P3</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_4"><i><font color="Green" title="E71">A78</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_4_1_1"/>
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_1_4_1_1"/>
<font color="Maroon" title="c19">P3</font> <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#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_4_1_2"/>
<font color="Maroon" title="c19">P3</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_1_4_1_2"/>
<font color="Maroon" title="c19">P3</font> <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#E19:37_5_2_1_1_7_1_1"><i><font color="Green" title="E67">A74</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></div><b>end;</b></div></div><b>end;</b></div><a NAME="E24:37_5_2_1_1_7_1_1"/><i><font color="Green" title="E71">A79</font></i>: 
<font color="Maroon" title="c15">Pmax</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_5_2_1_1_7_1_1"><i><font color="Green" title="E58">A63</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_1"><i><font color="Green" title="E59">A64</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_5_2_1_1_7_1_1"><i><font color="Green" title="E61">A66</font></i></a>, <a class="txt" href="taxonom2.html#E13:37_5_2_1_1_7_1_1"><i><font color="Green" title="E62">A67</font></i></a>, <a class="ref" href="taxonom2.html#T19" target="_self" title="TAXONOM2:th.19">Th19</a></i>;<br/><div><i><font color="Green" title="E72">A80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">P3</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_7_1_1_5"/><i><font color="Green" title="E73">A81</font></i>: 
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_1_5_1"/>
( <font color="Maroon" title="c12">Ca</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="c12">Ca</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_5_1_1"/>
<font color="Maroon" title="c12">Ca</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"><a NAME="E2:37_5_2_1_1_7_1_1_5_1_1"/><b>then </b>
<font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_5_2_1_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_5_2_1_1_7_1_1"><i><font color="Green" title="E56">A61</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_5"><i><font color="Green" title="E73">A81</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_5_2_1_1_7_1_1_5_1_1"/>
( <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E24:37_5_2_1_1_7_1_1"><i><font color="Green" title="E71">A79</font></i></a>, <a class="ref" href="setfam_1.html#T23" title="SETFAM_1:th.23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_5_1_2"/><i><font color="Green" title="E74">A82</font></i>: 
<font color="Maroon" title="c12">Ca</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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_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="E1:37_5_2_1_1_7_1_1_5_1_2_1_1"/>
( <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> or <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E13:37_5_2_1_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_5"><i><font color="Green" title="E73">A81</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green" title="E75">A83</font></i>: 
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c20">Pmin</font>, <font color="Maroon" title="c21">PY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green" title="E76">A84</font></i>: 
( <font color="Maroon" title="c20">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> &amp; <font color="Maroon" title="c21">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> )
 <b>and </b><br/><a NAME="E4:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green" title="E77">A85</font></i>: 
for <font color="Olive" title="b1">PZ</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> holds <br/>( <font color="Olive" title="b1">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">PY</font> &amp; <font color="Maroon" title="c20">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">PZ</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="taxonom2.html#E16:37_5_2_1_1"><i><font color="Green" title="E51">A54</font></i></a>, <a class="txt" href="taxonom2.html#E17:37_5_2_1_1"><i><font color="Green" title="E52">A55</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_5_1_2"><i><font color="Green" title="E74">A82</font></i></a></i>;<br/><a NAME="E5:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c20">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_5_2_1_1"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E76">A84</font></i></a></i>;<br/><a NAME="E6:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E78">A86</font></i>: 
<font color="Maroon" title="c20">Pmin</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">hw</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green" title="E79">A87</font></i>: 
( <font color="Maroon" title="c22">hw</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Pmin</font> &amp; <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">hw</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">hw</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_5_2_1_1"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E76">A84</font></i></a></i>;<br/><a NAME="E9:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c15">Pmax</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">Pmin</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1"/><i><font color="Green" title="E80">A88</font></i>: 
( <font color="Maroon" title="c20">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c15">Pmax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> )
 
<b>by </b><i><a class="txt" href="taxonom2.html#E13:37_5_2_1_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E76">A84</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E2:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E81">A89</font></i>: 
( <font color="Maroon" title="c20">Pmin</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c15">Pmax</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green" title="E82">A90</font></i>: 
<font color="Maroon" title="c20">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font>
 ;<br/><b>consider </b><font color="Maroon" title="c23">hx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green" title="E83">A91</font></i>: 
( <font color="Maroon" title="c23">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Pmin</font> &amp; <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">hx</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">hx</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_5_2_1_1"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E76">A84</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c24">hy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green" title="E84">A92</font></i>: 
( <font color="Maroon" title="c24">hy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Maroon" title="c24">hy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">hy</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_5_2_1_1"><i><font color="Green" title="E45">A39</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1"><i><font color="Green" title="E55">A60</font></i></a></i>;<br/><a NAME="E6:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green" title="E85">A93</font></i>: 
<font color="Maroon" title="c24">hy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">hx</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E83">A91</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E84">A92</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="c25">hz</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green" title="E86">A94</font></i>: 
( <font color="Maroon" title="c25">hz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Maroon" title="c23">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">hz</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E82">A90</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E83">A91</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/><a NAME="E9:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/>
not <font color="Maroon" title="c24">hy</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_5_2_1_1_7_1_1"><i><font color="Green" title="E58">A63</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E84">A92</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E10:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c24">hy</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c25">hz</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E85">A93</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E86">A94</font></i></a>, <a class="txt" href="taxonom2.html#E29"><i><font color="Green" title="E21">Lm4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c24">hy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">hz</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_5_2_1_1_7_1_1"><i><font color="Green" title="E58">A63</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E84">A92</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E86">A94</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E12:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E84">A92</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E86">A94</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E13:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green" title="E83">A91</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1"/>
<font color="Maroon" title="c15">Pmax</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">Pmin</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1"><i><font color="Green" title="E80">A88</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_5_2_1_1_7_1_1_5_1_2_1_1_1_1"><i><font color="Green" title="E81">A89</font></i></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E80">A95</font></i>: 
<font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">Pmin</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_5_2_1_1_7_1_1"><i><font color="Green" title="E58">A63</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_1"><i><font color="Green" title="E59">A64</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_5_2_1_1_7_1_1"><i><font color="Green" title="E61">A66</font></i></a>, <a class="txt" href="taxonom2.html#E13:37_5_2_1_1_7_1_1"><i><font color="Green" title="E62">A67</font></i></a>, <a class="txt" href="taxonom2.html#E6:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E78">A86</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E79">A87</font></i></a>, <a class="ref" href="taxonom2.html#T19" target="_self" title="TAXONOM2:th.19">Th19</a></i>;<br/><a NAME="E11:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c20">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E75">A83</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E77">A85</font></i></a></i>;<br/><b>hence </b><a NAME="E12:37_5_2_1_1_7_1_1_5_1_2_1_1_1"/>
( <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_5_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green" title="E80">A95</font></i></a>, <a class="ref" href="setfam_1.html#T23" title="SETFAM_1:th.23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_5_1_2_1_1_2"/>
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Cb</font>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_7_1_1_5_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37_5_2_1_1_7_1_1"><i><font color="Green" title="E56">A61</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_5_2_1_1_7_1_1_5_1_2_1_1_2"/>
( <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E24:37_5_2_1_1_7_1_1"><i><font color="Green" title="E71">A79</font></i></a>, <a class="ref" href="setfam_1.html#T23" title="SETFAM_1:th.23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:37_5_2_1_1_7_1_1_5_1_2"/>
( <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">P1</font>, <font color="Maroon" title="c20">P2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_7_1_1_6"/><i><font color="Green" title="E73">A96</font></i>: 
( <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_1_6_1"/>
( <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c19">P1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6"><i><font color="Green" title="E73">A96</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_6_1_1"/><i><font color="Green" title="E74">A97</font></i>: 
<font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_1_6_1_1_1"/>
( <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6"><i><font color="Green" title="E73">A96</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green" title="E75">A98</font></i>: 
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green" title="E76">A99</font></i>: 
<font color="Maroon" title="c19">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6_1_1"><i><font color="Green" title="E74">A97</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><a NAME="E3:37_5_2_1_1_7_1_1_6_1_1_1_1"/>
<font color="Maroon" title="c20">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6_1_1_1_1"><i><font color="Green" title="E75">A98</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_5_2_1_1_7_1_1_6_1_1_1_1"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6_1_1"><i><font color="Green" title="E74">A97</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6_1_1_1_1"><i><font color="Green" title="E75">A98</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_5_2_1_1_7_1_1_6_1_1_1_1"><i><font color="Green" title="E76">A99</font></i></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_6_1_1_1_2"/>
<font color="Maroon" title="c20">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_7_1_1_6_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <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="E3:37_5_2_1_1_7_1_1_6_1_1_1_2"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E25:37_5_2_1_1_7_1_1"><i><font color="Green" title="E72">A80</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6_1_1"><i><font color="Green" title="E74">A97</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_6_1_2"/>
<font color="Maroon" title="c19">P1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_7_1_1_6_1_2"/><b>then </b><i><font color="Green" title="E74">A100</font></i>: 
<font color="Maroon" title="c19">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Pb</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="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_1_6_1_2_1"/>
( <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_1_6"><i><font color="Green" title="E73">A96</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_2_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_6_1_2_1_1"/>
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_1_6_1_2_1_1"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E25:37_5_2_1_1_7_1_1"><i><font color="Green" title="E72">A80</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_5_2_1_1_7_1_1_6_1_2"><i><font color="Green" title="E74">A100</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_1_6_1_2_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_1_6_1_2_1_2"/>
<font color="Maroon" title="c20">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_1_6_1_2_1_2"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_5_2_1_1_7_1_1_6_1_2"><i><font color="Green" title="E74">A100</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></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E27:37_5_2_1_1_7_1_1"/><b>then </b>
<font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RL</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E22:37_5_2_1_1_7_1_1"><i><font color="Green" title="E69">A76</font></i></a>, <a class="txt" href="taxonom2.html#E23:37_5_2_1_1_7_1_1"><i><font color="Green" title="E70">A77</font></i></a></i>;<br/><a NAME="E28:37_5_2_1_1_7_1_1"/><b>then </b>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E21:37"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E29:37_5_2_1_1_7_1_1"/>
<font color="Maroon" title="c10">h</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="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E18:37_5_2_1_1_7_1_1"><i><font color="Green" title="E66">A73</font></i></a>, <a class="txt" href="taxonom2.html#E21:37_5_2_1_1_7_1_1"><i><font color="Green" title="E68">A75</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E55">A101</font></i>: 
<font color="Maroon" title="c13">Cb</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>then consider </b><font color="Maroon" title="c14">Pmin</font>, <font color="Maroon" title="c15">PY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E56">A102</font></i>: 
( <font color="Maroon" title="c14">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> &amp; <font color="Maroon" title="c15">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> )
 <b>and </b><br/><a NAME="E4:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E57">A103</font></i>: 
for <font color="Olive" title="b1">PZ</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ca</font> holds <br/>( <font color="Olive" title="b1">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">PY</font> &amp; <font color="Maroon" title="c14">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">PZ</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="taxonom2.html#E24:37"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="taxonom2.html#E13:37_5_2_1_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="txt" href="taxonom2.html#E17:37_5_2_1_1"><i><font color="Green" title="E52">A55</font></i></a></i>;<br/><a NAME="E5:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E58">A104</font></i>: 
<font color="Maroon" title="c14">Pmin</font> <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#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="taxonom2.html#E14:37_5_2_1_1"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_2"><i><font color="Green" title="E56">A102</font></i></a></i>;<br/><a NAME="E6:37_5_2_1_1_7_1_2"/>
<font color="Maroon" title="c14">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_5_2_1_1"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_2"><i><font color="Green" title="E56">A102</font></i></a></i>;<br/><a NAME="E7:37_5_2_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E59">A105</font></i>: 
<font color="Maroon" title="c14">Pmin</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">hw</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E60">A106</font></i>: 
( <font color="Maroon" title="c16">hw</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Pmin</font> &amp; <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">hw</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">hw</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_5_2_1_1"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_5_2_1_1_7_1_2"><i><font color="Green" title="E56">A102</font></i></a></i>;<br/><a NAME="E10:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E61">A107</font></i>: 
<font color="Maroon" title="c14">Pmin</font> <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="c16">hw</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#E5:37_5_2_1_1_7_1_2"><i><font color="Green" title="E58">A104</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">hw</font>;<br/><b>consider </b><font color="Maroon" title="c17">PT</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E62">A108</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">PT</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">PS</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</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>set </b><font color="Maroon" title="c18">PS1</font> = <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/><a NAME="E13:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E63">A109</font></i>: 
<font color="Maroon" title="c17">PT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E14:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E64">A110</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">PT</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_5_2_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_2"><i><font color="Green" title="E60">A106</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_5_2_1_1_7_1_2"><i><font color="Green" title="E62">A108</font></i></a></i>;<br/><a NAME="E15:37_5_2_1_1_7_1_2"/>
<font color="Maroon" title="c17">PT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">PS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">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="E1:37_5_2_1_1_7_1_2_1"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">PT</font>
 ;<br/>

<b>hence </b><a NAME="E2:37_5_2_1_1_7_1_2_1"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">PS</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E12:37_5_2_1_1_7_1_2"><i><font color="Green" title="E62">A108</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:37_5_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon" title="c17">PT</font> <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#E4:37_5_2_1_1"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:37_5_2_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E65">A111</font></i>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</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#E10:37_5_2_1_1_7_1_2"><i><font color="Green" title="E61">A107</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E18:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E66">A112</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">PS</font> or <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">hw</font> or <font color="Maroon" title="c16">hw</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> or <font color="Maroon" title="c16">hw</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_5_2_1_1"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1_7_1_2"><i><font color="Green" title="E58">A104</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_2"><i><font color="Green" title="E60">A106</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></i>;<br/><a NAME="E19:37_5_2_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E67">A113</font></i>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_5_2_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_5_2_1_1_7_1_2"><i><font color="Green" title="E59">A105</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_2"><i><font color="Green" title="E60">A106</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_5_2_1_1_7_1_2"><i><font color="Green" title="E62">A108</font></i></a>, <a class="ref" href="taxonom2.html#T18" target="_self" title="TAXONOM2:th.18">Th18</a></i>;<br/><a NAME="E20:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E68">A114</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">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="E1:37_5_2_1_1_7_1_2_2"/><i><font color="Green" title="E69">A115</font></i>: 
<font color="Maroon" title="c19">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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:37_5_2_1_1_7_1_2_2"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_2"><i><font color="Green" title="E69">A115</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_5_2_1_1_7_1_2_2"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E19:37_5_2_1_1_7_1_2"><i><font color="Green" title="E67">A113</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c19">C1</font> = <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E21:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E69">A116</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></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="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></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/><a NAME="E22:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E70">A117</font></i>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E23:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E71">A118</font></i>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">Pmin</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_5_2_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_5_2_1_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_5_2_1_1_7_1_2"><i><font color="Green" title="E59">A105</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_5_2_1_1_7_1_2"><i><font color="Green" title="E60">A106</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_5_2_1_1_7_1_2"><i><font color="Green" title="E62">A108</font></i></a>, <a class="txt" href="taxonom2.html#E18:37_5_2_1_1_7_1_2"><i><font color="Green" title="E66">A112</font></i></a>, <a class="ref" href="taxonom2.html#T18" target="_self" title="TAXONOM2:th.18">Th18</a></i>;<br/><a NAME="E24:37_5_2_1_1_7_1_2"/><i><font color="Green" title="E72">A119</font></i>: 
<font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E20:37_5_2_1_1_7_1_2"><i><font color="Green" title="E68">A114</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><div><i><font color="Green" title="E73">A120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">P3</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_7_1_2_3"/><i><font color="Green" title="E74">A121</font></i>: 
<font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_2_3_1"/>
( <font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P3</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_3"><i><font color="Green" title="E74">A121</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_3_1_1"/>
<font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_2_3_1_1"/>
<font color="Maroon" title="c20">P3</font> <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#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green" title="E34">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_3_1_2"/>
<font color="Maroon" title="c20">P3</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_2_3_1_2"/>
<font color="Maroon" title="c20">P3</font> <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#E17:37_5_2_1_1_7_1_2"><i><font color="Green" title="E65">A111</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></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green" title="E74">A122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">P3</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_7_1_2_4"/><i><font color="Green" title="E75">A123</font></i>: 
<font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><a NAME="E2:37_5_2_1_1_7_1_2_4"/>
<font color="Maroon" title="c14">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P3</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_5_2_1_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2"><i><font color="Green" title="E55">A101</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_5_2_1_1_7_1_2"><i><font color="Green" title="E57">A103</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_4"><i><font color="Green" title="E75">A123</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_5_2_1_1_7_1_2_4"/>
( <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P3</font> or <font color="Maroon" title="c20">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37_5_2_1_1_7_1_2"><i><font color="Green" title="E71">A118</font></i></a>, <a class="ref" href="setfam_1.html#T23" title="SETFAM_1:th.23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">P1</font>, <font color="Maroon" title="c21">P2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_5_2_1_1_7_1_2_5"/><i><font color="Green" title="E75">A124</font></i>: 
( <font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_2_5_1"/>
( <font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5"><i><font color="Green" title="E75">A124</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_5_1_1"/><i><font color="Green" title="E76">A125</font></i>: 
<font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_2_5_1_1_1"/>
( <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c21">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5"><i><font color="Green" title="E75">A124</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_5_1_1_1_1"/><i><font color="Green" title="E77">A126</font></i>: 
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_7_1_2_5_1_1_1_1"/><i><font color="Green" title="E78">A127</font></i>: 
<font color="Maroon" title="c20">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5_1_1"><i><font color="Green" title="E76">A125</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><a NAME="E3:37_5_2_1_1_7_1_2_5_1_1_1_1"/>
<font color="Maroon" title="c21">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5_1_1_1_1"><i><font color="Green" title="E77">A126</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_5_2_1_1_7_1_2_5_1_1_1_1"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5_1_1"><i><font color="Green" title="E76">A125</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5_1_1_1_1"><i><font color="Green" title="E77">A126</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_5_2_1_1_7_1_2_5_1_1_1_1"><i><font color="Green" title="E78">A127</font></i></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_5_1_1_1_2"/>
<font color="Maroon" title="c21">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_7_1_2_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <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="E3:37_5_2_1_1_7_1_2_5_1_1_1_2"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E26:37_5_2_1_1_7_1_2"><i><font color="Green" title="E74">A122</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5_1_1"><i><font color="Green" title="E76">A125</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_5_1_2"/>
<font color="Maroon" title="c20">P1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_5_2_1_1_7_1_2_5_1_2"/><b>then </b><i><font color="Green" title="E76">A128</font></i>: 
<font color="Maroon" title="c20">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_2_1_1_7_1_2_5_1_2_1"/>
( <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c21">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_5_2_1_1_7_1_2_5"><i><font color="Green" title="E75">A124</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_2_1_1"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_5_1_2_1_1"/>
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_2_5_1_2_1_1"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E26:37_5_2_1_1_7_1_2"><i><font color="Green" title="E74">A122</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_5_2_1_1_7_1_2_5_1_2"><i><font color="Green" title="E76">A128</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_1_7_1_2_5_1_2_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_1_7_1_2_5_1_2_1_2"/>
<font color="Maroon" title="c21">P2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_1_7_1_2_5_1_2_1_2"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_5_2_1_1_7_1_2_5_1_2"><i><font color="Green" title="E76">A128</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></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E28:37_5_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RL</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="taxonom2.html#E24:37_5_2_1_1_7_1_2"><i><font color="Green" title="E72">A119</font></i></a>, <a class="txt" href="taxonom2.html#E25:37_5_2_1_1_7_1_2"><i><font color="Green" title="E73">A120</font></i></a></i>;<br/><a NAME="E29:37_5_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E21:37"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E30:37_5_2_1_1_7_1_2"/>
<font color="Maroon" title="c10">h</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="c9">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_5_2_1_1_7_1_2"><i><font color="Green" title="E63">A109</font></i></a>, <a class="txt" href="taxonom2.html#E14:37_5_2_1_1_7_1_2"><i><font color="Green" title="E64">A110</font></i></a>, <a class="txt" href="taxonom2.html#E21:37_5_2_1_1_7_1_2"><i><font color="Green" title="E69">A116</font></i></a>, <a class="txt" href="taxonom2.html#E22:37_5_2_1_1_7_1_2"><i><font color="Green" title="E70">A117</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:37_5_2_1_1"/>
<font color="Maroon" title="c10">h</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="c9">C</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2_1_2"><b>suppose </b></a><a NAME="E1:37_5_2_1_2"/>
<font color="Maroon" title="c10">h</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="c9">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_5_2_1_2"/>
<font color="Maroon" title="c10">h</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="c9">C</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<b>hence </b><a NAME="E26:37"/>
( <font color="Maroon" title="c9">C</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green" title="E36">A25</font></i></a></i>;<br/>


</div>
