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<b>let </b><font color="Maroon" title="c1">X</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"> ( not <font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> iff  ex <font color="Olive" title="b1">X'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">X'</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )</span><br/>

<b>set </b><font color="Maroon" title="c2">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> ) </span>}</span> ;<br/>
<a NAME="E1:35"/><i><font color="Green" title="E14">A1</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="compact1.html#D9" target="_self" title="COMPACT1:def.9">Def9</a></i>;<br/>
<a NAME="E2:35"/><i><font color="Green" title="E15">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green" title="E16">A3</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="compact1.html#D9" target="_self" title="COMPACT1:def.9">Def9</a></i>;<br/>
<b>thus </b><a NAME="E4:35"/>
( not <font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> implies  ex <font color="Olive" title="b1">X'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">X'</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">X'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">X'</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> ) implies  not <font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/>
not <font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">X'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">X'</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/>

<a NAME="E2:35_1"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T10" title="COMPTS_1:th.10">COMPTS_1:10</a></i>;<br/>
<a NAME="E3:35_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="compact1.html#T4" target="_self" title="COMPACT1:th.4">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">X'</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">X'</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#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">X'</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/>

<b>thus </b><a NAME="E5:35_1"/>
<font color="Maroon" title="c3">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</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> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/>

<b>thus </b><a NAME="E6:35_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font></span><br/>

<a NAME="E7:35_1"/><i><font color="Green" title="E18">A5</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E8:35_1"/><i><font color="Green" title="E19">A6</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>reconsider </b><font color="Maroon" title="c4">Xe</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">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="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E3:35"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">XX</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">Xe</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/><span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xboole_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:35_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c5">XX</font>
<b>by </b><i><a class="txt" href="#E1:35_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T67" title="ZFMISC_1:th.67">ZFMISC_1:67</a></i>
;<br/><a NAME="E5:35_1_1"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c5">XX</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/><div><i><font color="Green" title="E22">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_2"/>
<font color="Maroon" title="c5">XX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_2"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:35_1_1_2"/>
contradiction
 ; <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><a NAME="E7:35_1_1"/>
<font color="Maroon" title="c5">XX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:35_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E8:35_1_1"/><b>then </b>
<font color="Maroon" title="c5">XX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E6:35_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E10:35_1_1"/><i><font color="Green" title="E23">A10</font></i>: 
<font color="Maroon" title="c5">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E11:35_1_1"/>
<font color="Maroon" title="c6">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E12:35_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c6">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_3"/>
<font color="Maroon" title="c6">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">XX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_1_3"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:35_1_1_3"/>
contradiction
 ; <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><a NAME="E14:35_1_1"/><b>then </b>
<font color="Maroon" title="c6">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E10:35_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T43" title="ZFMISC_1:th.43">ZFMISC_1:43</a></i>;<br/><b>hence </b><a NAME="E15:35_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E12:35_1_1"><i><font color="Green" title="E24">A11</font></i></a></i>; <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>hence </b><a NAME="E10:35_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font>
 <b>by </b><i><a class="txt" href="#E8:35_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>; <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>given </b><font color="Maroon" title="c3">X'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><a NAME="E6:35"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c3">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E7:35"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c3">X'</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a></span><br/>

<a NAME="E8:35"/><i><font color="Green" title="E19">A14</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:35"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E9:35"/>
<font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E10:35"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
;<br/>
<a NAME="E11:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
;<br/>
<a NAME="E12:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> where <font color="Olive" title="b1">U</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> ) </span>}</span> 
 
;<br/>
<a NAME="E13:35"/><b>then </b><i><font color="Green" title="E20">A15</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">X1</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"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E15:35"/>
<font color="Maroon" title="c4">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E13:35"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E16:35"/><b>then </b><i><font color="Green" title="E21">A16</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E17:35"/><i><font color="Green" title="E22">A17</font></i>: 
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<i><font color="Green" title="E23">A18</font></i>: <font color="Maroon" title="c4">X1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">X1</font>
<b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="txt" href="#E17:35"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>
;<br/>
<a NAME="E19:35"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compact1.html#K1" title="COMPACT1:func.1">One-Point_Compactification</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E3:35"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E20:35"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:35"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="#E8:35"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="#E16:35"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="#E18:35"><i><font color="Green" title="E23">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
