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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="abian.html#V2">covering</a> <a href="taxonom2.html#V5" target="_self">T_3</a>  <a href="taxonom2.html#M1" target="_self">Hierarchy</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="taxonom2.html#V6" target="_self">lower-bounded</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E25</font></i>: 
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>4</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>4</sub></font> ) ) )
 ;<br/>

<a NAME="E4:37"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="taxonom2.html#V3" target="_self">hierarchic</a>
 
<b>by </b><i><a class="ref" href="taxonom2.html#D4" target="_self">Def4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> ) );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E7:37"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="abian.html#T4">ABIAN:4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:37"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E13:37"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37"><i><font color="Green">E24</font></i></a>, <a class="txt" href="taxonom2.html#E2:37"><i><font color="Green">E25</font></i></a>, <a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E29</font></i></a>, <a class="ref" href="taxonom2.html#T17" target="_self">Th17</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E14:37"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></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">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:37_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:37"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D1">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">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:37_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:37_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E17:37"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E14:37"><i><font color="Green">E31</font></i></a>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E32</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">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:37_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:37_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/><div><i><font color="Green">E36</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">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:37_3_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:37_3_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E4:37_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_3_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_3_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:37_3_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_2"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37_3_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_3_2"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E38</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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>thus </b><a NAME="E4:37_3_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_3"><i><font color="Green">E39</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">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_4"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3_4"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_3_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_3_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D4">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">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E3:37_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3_4"><i><font color="Green">E41</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">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_3_4_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3_4"><i><font color="Green">E41</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">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_3_4_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_3_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3_4"><i><font color="Green">E41</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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_3"><i><font color="Green">E38</font></i></a></i>;<br/><b>thus </b><a NAME="E8:37_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:37"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E21:37"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> holds <br/> not <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E17:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_1.html#T175">ORDERS_1:175</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<a NAME="E23:37"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is    <a href="taxonom1.html#M1">Classification</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E2:37_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:37_4"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_4"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:37_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E3:37_5"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E17:37"><i><font color="Green">E33</font></i></a>, <a class="txt" href="taxonom2.html#E21:37"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_5"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:37"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_6"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:37_6_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37_6_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_6_1"><i><font color="Green">E39</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:37_6"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1"/>
(  not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:37_6_2_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:37_6_2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37"><i><font color="Green">E24</font></i></a>, <a class="txt" href="taxonom2.html#E2:37"><i><font color="Green">E25</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="taxonom2.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E5:37_6_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_6_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> );<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:37_6_2_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E8:37_6_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:37_6_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_6_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_1"><i><font color="Green">E45</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">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> );<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:37_6_2_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E11:37_6_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:37_6_2_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_6_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_2"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>assume </b><a NAME="E2:37_6_2_1_1_3"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:37_6_2_1_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:37_6_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:37_6_2_1_1_3"/>
<font color="Maroon">c<sub>14</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E6:37_6_2_1_1_3"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:37_6_2_1_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_6_2_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E9:37_6_2_1_1_3"/>
not <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37_6_2_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_6_2_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E10:37_6_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_6_2_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_6_2_1_1_3"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E13:37_6_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_6_2_1_1_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_4_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_4_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; not <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E12:37_6_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_4_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:37_6_2_1_1_4_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_4_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E5:37_6_2_1_1_4_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_4_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_4_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_4_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_4_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_6_2_1_1_4_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self">Def3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_6_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_4_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_4_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1_4_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_6_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_4_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_4_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1_4_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">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_6_2_1_1_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E8:37_6_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="taxonom2.html#E11:37_6_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:37_6_2_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E15:37_6_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_6_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E16:37_6_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_6_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_5"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><a NAME="E2:37_6_2_1_1_5"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_6_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_5"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E3:37_6_2_1_1_5"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_5"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taxonom2.html#E14:37_6_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_5"><i><font color="Green">E53</font></i></a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E18:37_6_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_6_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_6"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E2:37_6_2_1_1_6"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_6_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_6"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E3:37_6_2_1_1_6"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_6"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taxonom2.html#E15:37_6_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_6"><i><font color="Green">E55</font></i></a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>and </b><br/><a NAME="E4:37_6_2_1_1_7_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="taxonom2.html#E18:37_6_2_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="taxonom2.html#E19:37_6_2_1_1"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E5:37_6_2_1_1_7_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E15:37_6_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_6_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E7:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_6_2_1_1_7_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E10:37_6_2_1_1_7_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_1"><i><font color="Green">E59</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:37_6_2_1_1_7_1_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E13:37_6_2_1_1_7_1_1"/><i><font color="Green">E62</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_6_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E14:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_6_2_1_1_7_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_6_2_1_1_7_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="taxonom2.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E15:37_6_2_1_1_7_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:37_6_2_1_1_7_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_6_2_1_1_7_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:37_6_2_1_1_7_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:37_6_2_1_1_7_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:37_6_2_1_1_7_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E18:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E19:37_6_2_1_1_7_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E11:37_6_2_1_1_7_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taxonom2.html#E16:37_6_2_1_1_7_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E20:37_6_2_1_1_7_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E21:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E22:37_6_2_1_1_7_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E15:37_6_2_1_1_7_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_4"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_1_4_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_4_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_4_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_4_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_4_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E19:37_6_2_1_1_7_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tarski.html#D1">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_6_2_1_1_7_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_6_2_1_1_7_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_6_2_1_1_7_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="taxonom2.html#E13:37_6_2_1_1_7_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="taxonom2.html#T19" target="_self">Th19</a></i>;<br/><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_5"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_1_5_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_5_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_6_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_6_2_1_1_7_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_5"><i><font color="Green">E73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_5_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E24:37_6_2_1_1_7_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="setfam_1.html#T23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_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_6_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_6_2_1_1_7_1_1_5_1_2_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E13:37_6_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_5"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E4:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="taxonom2.html#E16:37_6_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="taxonom2.html#E17:37_6_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_5_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_6_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_6_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E9:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="taxonom2.html#E13:37_6_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E2:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_6_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_6_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><a NAME="E9:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/>
not <font color="Maroon">c<sub>24</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_6_2_1_1_7_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E11:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="taxonom2.html#E29"><i><font color="Green">Lemma21</font></i></a></i>;<br/><a NAME="E12:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_6_2_1_1_7_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E13:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E14:37_6_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_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_6_2_1_1_7_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_6_2_1_1_7_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="taxonom2.html#E13:37_6_2_1_1_7_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="taxonom2.html#E6:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="taxonom2.html#E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="taxonom2.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E11:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E77</font></i></a></i>;<br/><b>hence </b><a NAME="E12:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_6_2_1_1_7_1_1_5_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="setfam_1.html#T23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_5_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37_6_2_1_1_7_1_1"><i><font color="Green">E56</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_5_1_2_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E24:37_6_2_1_1_7_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="setfam_1.html#T23">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_6_2_1_1_7_1_1_5_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</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_6_2_1_1_7_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_6"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_1_6_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_1_6_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E3:37_6_2_1_1_7_1_1_6_1_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_7_1_1_6_1_1_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_6_2_1_1_7_1_1_6_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_6_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_6_1_1_1_2"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E25:37_6_2_1_1_7_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6_1_1"><i><font color="Green">E74</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_6_2_1_1_7_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_6_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_1_6_1_2_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_1_6"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_2_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_2_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_6_1_2_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E25:37_6_2_1_1_7_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_6_2_1_1_7_1_1_6_1_2"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_2_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_2_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_6_1_2_1_2"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_6_2_1_1_7_1_1_6_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tarski.html#D1">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_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E22:37_6_2_1_1_7_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="taxonom2.html#E23:37_6_2_1_1_7_1_1"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E28:37_6_2_1_1_7_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E29:37_6_2_1_1_7_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E21:37"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taxonom2.html#E27:37_6_2_1_1_7_1_1"><i><font color="Green">E73</font></i></a></i>;<br/><b>hence </b><a NAME="E30:37_6_2_1_1_7_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E18:37_6_2_1_1_7_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="taxonom2.html#E21:37_6_2_1_1_7_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E4:37_6_2_1_1_7_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="taxonom2.html#E24:37"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taxonom2.html#E13:37_6_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E17:37_6_2_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E5:37_6_2_1_1_7_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E14:37_6_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E14:37_6_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E7:37_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_6_2_1_1_7_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:37_6_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taxonom2.html#E3:37_6_2_1_1_7_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E10:37_6_2_1_1_7_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:37_6_2_1_1_7_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/><a NAME="E13:37_6_2_1_1_7_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E14:37_6_2_1_1_7_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_6_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_6_2_1_1_7_1_2"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E15:37_6_2_1_1_7_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E12:37_6_2_1_1_7_1_2"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:37_6_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37_6_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:37_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:37_6_2_1_1_7_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E18:37_6_2_1_1_7_1_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_6_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1_7_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E19:37_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_6_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_6_2_1_1_7_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_6_2_1_1_7_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="taxonom2.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E20:37_6_2_1_1_7_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:37_6_2_1_1_7_1_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_6_2_1_1_7_1_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E19:37_6_2_1_1_7_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E21:37_6_2_1_1_7_1_2"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E22:37_6_2_1_1_7_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E23:37_6_2_1_1_7_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:37_6_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom2.html#E5:37_6_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="taxonom2.html#E7:37_6_2_1_1_7_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="taxonom2.html#E9:37_6_2_1_1_7_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="taxonom2.html#E12:37_6_2_1_1_7_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="taxonom2.html#E18:37_6_2_1_1_7_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="taxonom2.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E24:37_6_2_1_1_7_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E20:37_6_2_1_1_7_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_2_3_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_3_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_3_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_3_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_3_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E17:37_6_2_1_1_7_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D1">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">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:37_6_2_1_1_7_1_2_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_6_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="taxonom2.html#E4:37_6_2_1_1_7_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_4"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_2_4"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37_6_2_1_1_7_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="setfam_1.html#T23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_5"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_2_5_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_2_5_1_1_1"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_2_5_1_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E3:37_6_2_1_1_7_1_2_5_1_1_1_1"/>
<font color="Maroon">c<sub>21</sub></font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_7_1_2_5_1_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green">E36</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_6_2_1_1_7_1_2_5_1_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_1_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_2_5_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_2_5_1_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E26:37_6_2_1_1_7_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5_1_1"><i><font color="Green">E76</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_6_2_1_1_7_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_2_5_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_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_6_2_1_1_7_1_2_5_1_2_1"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:37_6_2_1_1_7_1_2_5"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_2_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_2_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_5_1_2_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E26:37_6_2_1_1_7_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="taxonom2.html#E2:37_6_2_1_1_7_1_2_5_1_2"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_2_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_2_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_5_1_2_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:37_6_2_1_1_7_1_2_5_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="tarski.html#D1">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_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="taxonom2.html#E24:37_6_2_1_1_7_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="taxonom2.html#E25:37_6_2_1_1_7_1_2"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E29:37_6_2_1_1_7_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E30:37_6_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E21:37"><i><font color="Green">E35</font></i></a>, <a class="txt" href="taxonom2.html#E28:37_6_2_1_1_7_1_2"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E31:37_6_2_1_1_7_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37_6_2_1_1_7_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="taxonom2.html#E14:37_6_2_1_1_7_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="taxonom2.html#E21:37_6_2_1_1_7_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="taxonom2.html#E22:37_6_2_1_1_7_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="tarski.html#D4">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_6_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> is    <a href="taxonom1.html#M1">Classification</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E23:37"><i><font color="Green">E36</font></i></a></i>;<br/>


</div>
