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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">PA</font>, <font color="Maroon" title="c3">PB</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#NR2" title="PARTIT1:NR.2">'&gt;'</a> <font color="Maroon" title="c3">PB</font>
 ;<br/>

<a NAME="E2:6"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">x</font>;<br/>
<a NAME="E4:6_1"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E2:6_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<a NAME="E5:6_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">PB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:6_1"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E1:6_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="partit1.html#E4:6_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:6_1"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">z</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="partit1.html#E7:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/>
<a NAME="E10:6_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> or <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">z</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E1:6_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="partit1.html#E9:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E11:6_1"/>
 ex <font color="Olive" title="b1">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E2:6_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="partit1.html#E7:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="partit1.html#E9:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
<font color="Maroon" title="c2">PA</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c3">PB</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D3" title="SETFAM_1:def.3">SETFAM_1:def 3</a></i>;<br/>


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