<?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>let </b><font color="Maroon" title="c3">X0</font>, <font color="Maroon" title="c4">X1</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c3">X0</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c4">X1</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c3">X0</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">X1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c5">A</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>  &amp; <font color="Maroon" title="c3">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:12"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c6">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>  &amp; <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X1</font>
 ;<br/>

<a NAME="E2:12_1_1"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</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="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:12_1_1"/><i><font color="Green" title="E18">A7</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E3:12"><i><font color="Green" title="E13">A2</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="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:12_1_1"/><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">z</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E5:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="partit1.html#E2:12_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:12_1_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E3:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="partit1.html#E4:12_1_1"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/>
<a NAME="E8:12_1_1"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="partit1.html#E6:12_1_1"><i><font color="Green" title="E19">A8</font></i></a></i>;<br/>
<a NAME="E9:12_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">z</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E4:12_1_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="partit1.html#E6:12_1_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E10:12_1_1"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E7:12_1_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="partit1.html#E8:12_1_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E11:12_1_1"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E4:12_1_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="partit1.html#E6:12_1_1"><i><font color="Green" title="E19">A8</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="E12:12_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="partit1.html#E4:12_1_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:12_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_1_2"/><i><font color="Green" title="E17">A11</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:12_1_2"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E3:12_1_2"><i><font color="Green" title="E17">A11</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="E5:12_1_2"/><b>then </b>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E3:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="partit1.html#E5:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="partit1.html#E3:12_1_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:12_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X1</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="partit1.html#E1:12_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:12"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E3:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="partit1.html#E5:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green" title="E17">A13</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E9:12"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E8:12"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:12"/><b>then </b><i><font color="Green" title="E18">A14</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E7:12"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E20">A16</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>thus </b><a NAME="E4:12_2"/>
contradiction
 <b>by </b><i><a class="txt" href="partit1.html#E6:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="partit1.html#E1:12_2"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="partit1.html#E3:12_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:12"/>
<font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X1</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 <b>by </b><i><a class="txt" href="partit1.html#E6:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="partit1.html#E11:12"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></i>;<br/>


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