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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a>  implies ( <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a>  iff for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">Y</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="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a>  iff for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">Y</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="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">Y</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="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  ) ) implies <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">Y</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="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Y</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/><a NAME="E2:56_1"/>
 ex <font color="Olive" title="b1">Y</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b1">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="waybel_3.html#D9" title="WAYBEL_3:def.9">WAYBEL_3:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:56_1"/>
 ex <font color="Olive" title="b1">Y</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:56"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">Y</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="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V6" title="WAYBEL_3:attr.6">locally-compact</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_3.html#D9" title="WAYBEL_3:def.9">WAYBEL_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>  holds <br/>(  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  ) )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  not <font color="Maroon" title="c3">X</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E4:56"/>
( <font color="Maroon" title="c2">x</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>

<a NAME="E5:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E7:56"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">and </span><br/><a NAME="E8:56"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c4">Y</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>;<br/>
<a NAME="E9:56"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:56"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T46" title="TOPS_1:th.46">TOPS_1:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E12:56"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c5">Q</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E13:56"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">and </span><br/><a NAME="E14:56"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="yellow_8.html#T36" target="_self" title="YELLOW_8:th.36">Th36</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">Q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E15:56"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56"><span class="lab"><font color="Green" title="E35">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>

<a NAME="E16:56"/>
<font color="Maroon" title="c3">X</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:56"/>
<font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<a NAME="E18:56"/>
<font color="Maroon" title="c3">X</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:56"/>
<font color="Maroon" title="c5">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E12:56"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="compts_1.html#T18" title="COMPTS_1:th.18">COMPTS_1:18</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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