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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 ;<br/>

<a NAME="E3:67"/><i><font color="Green" title="E43">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">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" title="E44">A4</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a>
 ;<br/><b>thus </b><a NAME="E2:67_1"/>
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">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" title="E45">A5</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:67"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="waybel12.html#E2:67"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="yellow_1.html#T24" title="YELLOW_1:th.24">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" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">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/><a NAME="E2:67_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">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#E1:67_1_1"><i><font color="Green" title="E45">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="E3:67_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c3">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> 
 
;<br/>
<b>hence </b><a NAME="E4:67_1_1"/>
not <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D4" title="YELLOW_8:def.4">YELLOW_8:def 4</a><br/>

<b>thus </b><a NAME="E5:67_1_1"/>
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">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" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:67"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="waybel12.html#E3:67"><i><font color="Green" title="E43">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a>,<a class="ref" href="waybel12.html#D1" target="_self" title="WAYBEL12:def.1">WAYBEL12:def 1</a><br/>


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

<b>let </b><font color="Maroon" title="c4">S1</font>, <font color="Maroon" title="c5">S2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E6:67_1_1"/><i><font color="Green" title="E46">A6</font></i>: 
( <font color="Maroon" title="c4">S1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">S2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>and </b><br/><a NAME="E7:67_1_1"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">S2</font>
 ;<br/>

<a NAME="E8:67_1_1"/><i><font color="Green" title="E48">A8</font></i>: 
( <font color="Maroon" title="c4">S1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">S2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E6:67_1_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">s1</font> = <font color="Maroon" title="c4">S1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> , <font color="Maroon" title="c7">s2</font> = <font color="Maroon" title="c5">S2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="txt" href="waybel12.html#E3:67"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:67_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel12.html#E8:67_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>
<a NAME="E11:67_1_1"/><b>then </b><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c6">s1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<font color="Maroon" title="c3">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c5">S2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="waybel12.html#E7:67_1_1"><i><font color="Green" title="E47">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green" title="E6">Lm2</font></i></a></i>
;<br/>
<a NAME="E13:67_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">s2</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:67"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="waybel12.html#E11:67_1_1"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="yellow_1.html#T9" title="YELLOW_1:th.9">YELLOW_1:9</a></i>;<br/>
<a NAME="E14:67_1_1"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font> or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:67_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="waybel_6.html#D2" title="WAYBEL_6:def.2">WAYBEL_6:def 2</a></i>;<br/>
<b>hence </b><a NAME="E15:67_1_1"/>
( <font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or <font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="txt" href="waybel12.html#E1:67"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E5:67"/>
( <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a> implies <font color="Maroon" title="c2">A</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a> )
 <b>by </b><i><a class="txt" href="waybel12.html#E1:67"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="waybel12.html#T44" target="_self" title="WAYBEL12:th.44">Th44</a></i>;<br/>


</div>
