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<b>let </b><font color="Maroon">X</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">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<a NAME="E1:38"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> <font color="Maroon">f</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> <font color="Maroon">f</font> &amp; not <font color="Olive">A</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> <font color="Maroon">f</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">a1</font> being  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">R1</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Maroon">a1</font></span>)</span> <a href="lang1.html#K17">[*]</a> 
 <b>and </b><br/><a NAME="E4:38_1"/><i><font color="Green">E31</font></i>: 
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:38_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">a2</font> being  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">R2</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:38_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Maroon">a2</font></span>)</span> <a href="lang1.html#K17">[*]</a> 
 <b>and </b><br/><a NAME="E7:38_1"/><i><font color="Green">E37</font></i>: 
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="taxonom1.html#T2" target="_self">Th2</a>, <a class="txt" href="taxonom1.html#E3:38_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">R1</font>,<font color="Maroon">c</font> )
 <b>by </b><i>, <a class="txt" href="taxonom1.html#E1:38_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">R2</font>,<font color="Maroon">c</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E6:38_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E7:38_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="taxonom1.html#E3:38_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom1.html#E5:38_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>thus </b><a NAME="E7:38_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="txt" href="taxonom1.html#E1:38_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="taxonom1.html#E3:38_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom1.html#E5:38_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">R2</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="taxonom1.html#E7:38_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_1_1_1_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">R1</font>,<font color="Maroon">c</font>
 ;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E6:38_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>thus </b><a NAME="E7:38_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_2"/>
( <font color="Maroon">A</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1"/>
( <font color="Maroon">A</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">A</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:38"/>
 <a href="taxonom1.html#K2" target="_self">fam_class</a> <font color="Maroon">f</font> is    <a href="taxonom1.html#M1" target="_self">Classification</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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