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

<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> 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>thus </b><a NAME="E1:48"/>
<span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partit1.html#T27" target="_self" title="PARTIT1:th.27">Th27</a></i>;<br/>
<a NAME="E2:48_1"/>
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T37" target="_self" title="PARTIT1:th.37">Th37</a></i>;<br/>
<a NAME="E3:48_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T1" title="EQREL_1:th.1">EQREL_1:1</a></i>;<br/>
<a NAME="E4:48_1"/><b>then </b>
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E5:48_1"/><b>then </b><i><font color="Green" title="E32">A2</font></i>: 
 <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E1:48_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="partit1.html#T24" target="_self" title="PARTIT1:th.24">Th24</a></i>;<br/>
<a NAME="E6:48_1"/>
 <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font> <a href="partit1.html#NR2" title="PARTIT1:NR.2">'&gt;'</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partit1.html#T36" target="_self" title="PARTIT1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1"/>
<span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="partit1.html#E5:48_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="partit1.html#T5" target="_self" title="PARTIT1:th.5">Th5</a></i>;<br/>


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

<b>thus </b><a NAME="E2:48"/>
<span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c2">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">PA</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<a NAME="E1:48_2"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partit1.html#T28" target="_self" title="PARTIT1:th.28">Th28</a></i>;<br/>
<a NAME="E2:48_2"/>
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T37" target="_self" title="PARTIT1:th.37">Th37</a></i>;<br/>
<a NAME="E3:48_2"/><b>then </b>
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E1:48_2"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E4:48_2"/>
<span class="p1">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c2">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">PA</font>
 <b>by </b><i><a class="ref" href="partit1.html#T29" target="_self" title="PARTIT1:th.29">Th29</a></i>;<br/>


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


</div>
