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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">S</font> be  <a href="yellow_8.html#V2">irreducible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">V</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>assume </b><a NAME="E1:19"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:19"/><i><font color="Green">E37</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/>
<a NAME="E3:19"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D4">YELLOW_8:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">sL</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>;<br/>
<b>let </b><font color="Maroon">X</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>, <font color="Maroon">Y</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D6">WAYBEL_6:def 6</a><br/>

<b>assume </b><a NAME="E4:19"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">Y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E5:19"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Y</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/>;<br/>
<b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font>, <font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E7:19"/>
( <font color="Maroon">X'</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y'</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:19"/><b>then </b>
( <font color="Maroon">X'</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">Y'</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E9:19"/><b>then </b><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">Y'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E10:19"/>
<font color="Maroon">X'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y'</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#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E11:19"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T9">YELLOW_1:9</a></i>;<br/>
<a NAME="E12:19"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E13:19"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y'</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:19"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y'</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/>
<a NAME="E15:19"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Y'</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E16:19"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E17:19"/><b>then </b>
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> or <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y'</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_8.html#D4">YELLOW_8:def 4</a></i>;<br/>
<a NAME="E18:19"/><b>then </b>
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X'</font> <a href="subset_1.html#K3">`</a>  or <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y'</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E19:19"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> or <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E20:19"/>
( <font color="Maroon">X</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">V</font> or <font color="Maroon">Y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


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