<?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>, <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/>



<a NAME="E1:33"/><i><font color="Green" title="E26">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="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c3">PB</font> <a href="partit1.html#NR2" title="PARTIT1:NR.2">'&gt;'</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font> )
 
<b>by </b><i><a class="ref" href="partit1.html#T17" target="_self" title="PARTIT1:th.17">Th17</a></i>;<br/>
<a NAME="E2:33"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font></span>)</span> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="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="c3">PB</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c7">A</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="partit1.html#E3:33_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:33_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="partit1.html#E3:33_1_1"><i><font color="Green" title="E27">A2</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="E8:33_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E3:33_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="partit1.html#E5:33_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="partit1.html#E7:33_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E9:33_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="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="c3">PB</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:33_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="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="c3">PB</font></span>)</span>
 ;<br/>

<a NAME="E3:33_1"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E5:33_1"/><i><font color="Green" title="E28">A6</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E7:33_1"/><i><font color="Green" title="E29">A7</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E3:33_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></i>;<br/>
<a NAME="E8:33_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="partit1.html#E5:33_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="partit1.html#E7:33_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E10:33_1"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">B</font>
 ;<br/>
<a NAME="E11:33_1"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:33_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="partit1.html#E7:33_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="partit1.html#E10:33_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E12:33_1"/>
not <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="partit1.html#E8:33_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="partit1.html#E10:33_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:33_1"/><b>then </b><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="partit1.html#E11:33_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:33_1"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:33_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="partit1.html#E7:33_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="partit1.html#E10:33_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:33_1"/>
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:33_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="partit1.html#E7:33_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="partit1.html#E10:33_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:33_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 <b>by </b><i><a class="txt" href="partit1.html#E13:33_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="partit1.html#E14:33_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</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> <font color="Maroon" title="c2">PA</font></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="c3">PB</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></i>;<br/>


</div>
