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

<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">D</font> be  <a href="tops_2.html#V1">open</a> <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font> is <a href="connsp_1.html#V1">connected</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/>

<b>assume </b><a NAME="E4:16"/>
<font color="Maroon">D</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E5:16"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:16"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:16"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E9:16"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</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="E11:16"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E12:16"/><i><font color="Green">E67</font></i>: 
<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="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>set </b><font color="Maroon">D1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">D2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span> ;<br/>
<a NAME="E13:16"/><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E14:16"/><i><font color="Green">E69</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E15:16"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:16_1"/><b>then </b>
 ex <font color="Olive">K1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">K1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Olive">K1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K1</font> )
 
;<br/>
<a NAME="E3:16_1"/><b>then </b>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E17:16"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:16_2"/><b>then </b>
 ex <font color="Olive">K1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">K1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Olive">K1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K1</font> )
 
;<br/>
<a NAME="E3:16_2"/><b>then </b>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E19:16"/>
<font color="Maroon">D1</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:16_3"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">K1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:16_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:16_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">K1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E5:16_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 ;<br/>
<a NAME="E6:16_3"/>
<font color="Maroon">K1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:16_3"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:16"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>
<a NAME="E21:16"/>
<font color="Maroon">D2</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:16_4"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">K1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:16_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:16_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">K1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E5:16_4"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 ;<br/>
<a NAME="E6:16_4"/>
<font color="Maroon">K1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:16_4"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:16"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>
<a NAME="E23:16"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E24:16"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E25:16"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_5"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_5_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_5_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:16_5_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E6:16_5_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_5_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_5_1_1_1"><b>suppose </b></a><a NAME="E1:16_5_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><div class="add"><a NAME="E2:16_5_1_1_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 <b>by </b><i/>;<br/><a NAME="E3:16_5_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:16_5_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_5_1_1_2"><b>suppose </b></a><a NAME="E1:16_5_1_1_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><div class="add"><a NAME="E2:16_5_1_1_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i/>;<br/><a NAME="E3:16_5_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:16_5_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:16_5"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E26:16"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E27:16"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E28:16"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E29:16"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, , , , , , <a class="ref" href="connsp_1.html#T12">CONNSP_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E31:16"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D1</font>
 <b>and </b><br/><a NAME="E32:16"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E34:16"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E35:16"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E37:16"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>and </b><br/><a NAME="E38:16"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">K1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E40:16"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E41:16"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">K1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E42:16"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">K2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E44:16"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E45:16"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">K2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E46:16"/><i><font color="Green">E96</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font>
 <b>by </b><i/>;<br/>
<a NAME="E47:16"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> )
 
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E48:16"/><b>then </b>
<font color="Maroon">K1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">K2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E49:16"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="taxonom2.html#D5">TAXONOM2:def 5</a></i>;<br/>


</div>
