<?xml version="1.0"?>
<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>  &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</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><br/>

<span class="kw">set </span><font color="Maroon" title="c2">L</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E1:74"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#D1" title="YELLOW_1:def.1">YELLOW_1:def 1</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:74"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a> 
 <span class="kw">and </span><br/><a NAME="E3:74"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</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="c3">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" target="_self" 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" 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>  st <font color="Maroon" title="c3">x</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>  holds <br/> 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="Maroon" title="c3">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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</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="c4">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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c4">X</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  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="c3">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> <font color="Maroon" title="c4">X</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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:74"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E5:74"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c4">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" 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="c3">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> <font color="Maroon" title="c4">X</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">X</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_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:74"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<font color="Maroon" title="c5">a</font> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="waybel_3.html#D5" target="_self" title="WAYBEL_3:def.5">Def5</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:74"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <span class="kw">and </span><br/><a NAME="E10:74"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E11:74"/>
<font color="Maroon" title="c6">Y</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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:74"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c6">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:74"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">and </span><br/><a NAME="E15:74"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:74"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>;<br/>
<a NAME="E16:74"/>
<font color="Maroon" title="c7">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E10:74"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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="E18:74"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">and </span><br/><a NAME="E19:74"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E9:74"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="urysohn1.html#T29" title="URYSOHN1:th.29">URYSOHN1:29</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c10">Z</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">W</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="c3">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="c10">Z</font> &amp; <font color="Maroon" title="c10">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="c10">Z</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>

<a NAME="E20:74"/>
<font color="Maroon" title="c9">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:74"/>
<font color="Maroon" title="c3">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="c10">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:74"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</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="c10">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="c10">Z</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>

<a NAME="E22:74"/>
<font color="Maroon" title="c8">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:74"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="waybel_3.html#T1" target="_self" title="WAYBEL_3:th.1">Th1</a></span>;<br/>
<a NAME="E23:74"/><span class="kw">then </span>
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:74"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:74"/>
<font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:74"><span class="lab"><font color="Green" title="E42">A11</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="c10">Z</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="c11">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a> <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="c11">F</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c10">Z</font> or  not <font color="Maroon" title="c11">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">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">K10</a>(<a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</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="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F</font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c10">Z</font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E25:74"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon" title="c11">F</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c10">Z</font>
 <span class="kw">and </span><br/><a NAME="E26:74"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c11">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">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">K10</a>(<a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</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="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F</font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c10">Z</font> &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">ZZ</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">ZZ</font> = <font color="Maroon" title="c12">ZZ</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">FZ</font> = <font color="Maroon" title="c11">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c13">ZZ</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E30:74"/>
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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">ZZ</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E31:74"/><span class="kw">then </span>
<font color="Maroon" title="c13">ZZ</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<a NAME="E32:74"/><span class="kw">then </span>
<font color="Maroon" title="c14">FZ</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:74"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="tops_2.html#T20" title="TOPS_2:th.20">TOPS_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">FF</font> = <font color="Maroon" title="c14">FZ</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T25" title="YELLOW_1:th.25">YELLOW_1:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E45">A14</font></span>:  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font> = 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c14">FZ</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">ZZ</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>
;<br/>
<a NAME="E35:74"/>
<font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c11">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:74"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/>
<a NAME="E36:74"/><span class="kw">then </span>
<font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:74"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></span>;<br/>
<a NAME="E37:74"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/>
<a NAME="E38:74"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E39:74"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E41:74"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c16">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">FF</font>
 <span class="kw">and </span><br/><a NAME="E42:74"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:74"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="waybel_3.html#T18" target="_self" title="WAYBEL_3:th.18">Th18</a></span>;<br/>
<a NAME="E43:74"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">G</font> = <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">ZZ</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">G</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="c17">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F</font> &amp; <font color="Maroon" title="c17">G</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c17">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E45:74"/>
<font color="Maroon" title="c17">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:74"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</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="c17">G</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c17">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E46:74"/>
<font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">G</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</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="c17">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:74_3"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</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="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">G</font></span><br/>

<a NAME="E2:74_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:74"><span class="lab"><font color="Green" title="E42">A11</font></span></a></span>;<br/>
<a NAME="E3:74_3"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:74"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E42:74"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E43:74"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E4:74_3"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
not <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_3"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:74_3"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font>
 <span class="kw">and </span><br/><a NAME="E7:74_3"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_3"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E3:74_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E8:74_3"/>
not <font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">ZZ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_3"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E6:74_3"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:74_3"/><span class="kw">then </span>
<font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:74_3"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:74_3"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_3"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E47:74"/>
<font color="Maroon" title="c17">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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>
