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<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:91_1_1"/>
 <a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" target="_self" title="YELLOW_6:def.5">YELLOW_6:def 5</a><br/>

<b>consider </b><font color="Maroon" title="c6">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:91_1_1"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c6">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:91_1_1"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">W</font>;<br/>
<a NAME="E6:91_1_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:91_1_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="yellow_6.html#E5:91_1_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:91_1_1"/><b>then </b>
<font color="Maroon" title="c6">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">W</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T12" target="_self" title="YELLOW_6:th.12">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">z</font>
;<br/>

<a NAME="E9:91_1_1"/>
( <font color="Maroon" title="c9">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c9">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:91_1_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="yellow_6.html#E5:91_1_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E10:91_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font> &amp; <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T40" target="_self" title="YELLOW_6:th.40">Th40</a></i>;<br/>


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