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<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:91_1_1"/>
 <a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font> is <a href="orders_2.html#V3">transitive</a>
 ;<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_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</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_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" target="_self">YELLOW_6:def 5</a><br/>

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:91_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:91_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font>;<br/>
<a NAME="E6:91_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font> &amp; <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font> 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>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:91_1_1"/><b>then </b>
<font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 
;<br/>
<a NAME="E8:91_1_1"/><b>then </b>
<font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> </span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E10:91_1_1"/>
( <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E11:91_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D4" target="_self">Def4</a></i>;<br/>


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