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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Olive" title="b1">p</font></span>)</span> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b2">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> iff ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> iff ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )</span><br/>





<b>set </b><font color="Maroon" title="c4">Xp</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>reconsider </b><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>thus </b><a NAME="E2:89"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> implies ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<a NAME="E2:89_1"/><b>then </b>
 ex <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <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> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E3:89_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><b>that </b><br/><a NAME="E3:89"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 <b>and </b><br/><a NAME="E4:89"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c3">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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></span><br/>

<a NAME="E5:89"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> <span class="p2">{<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> </span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:89"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="yellow_6.html#E4:89"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:89"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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