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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 ;<br/>

<a NAME="E3:67"/><i><font color="Green">E28</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> is <a href="waybel_6.html#V2">meet-irreducible</a>
 ;<br/><b>thus </b><a NAME="E2:67_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  is <a href="yellow_8.html#V2">irreducible</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:67_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="waybel12.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow_1.html#T24">YELLOW_1:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:67_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E3:67_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:67_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>hence </b><a NAME="E4:67_1_1"/>
not <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D4">YELLOW_8:def 4</a><br/>

<b>thus </b><a NAME="E5:67_1_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:67_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T4" target="_self">Th4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>,<a class="ref" href="waybel12.html#D1" target="_self">WAYBEL12:def 1</a><br/>


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

<b>let </b><font color="Maroon">S1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">S2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E6:67_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">S1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">S2</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E7:67_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">S2</font>
 ;<br/>

<a NAME="E8:67_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">S1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">S2</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">S1</font> <a href="subset_1.html#K3">`</a> , <font color="Maroon">s2</font> = <font color="Maroon">S2</font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="waybel12.html#T4" target="_self">Th4</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:67_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E11:67_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">s1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">S2</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E13:67_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="waybel12.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>
<a NAME="E14:67_1_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="waybel_6.html#D2">WAYBEL_6:def 2</a></i>;<br/>
<b>hence </b><a NAME="E15:67_1_1"/>
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  or <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:67"/>
( <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  is <a href="yellow_8.html#V2">irreducible</a> implies <font color="Maroon">A</font> is <a href="waybel_6.html#V2">meet-irreducible</a> )
 <b>by </b><i>, </i>;<br/>


</div>
