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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a>  holds <br/><font color="Olive" title="b1">A</font> is  <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Maroon" title="c2">A</font> is  <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">A</font> is  <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a>  or <font color="Maroon" title="c2">A</font> is  <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:77"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> is  <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:77"/>
 not <font color="Maroon" title="c2">A</font> is  <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:77"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c3">D</font> is  <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a> 
 <span class="kw">and </span><br/><a NAME="E6:77"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">and </span><br/><a NAME="E7:77"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="tex_2.html#D5" target="_self" title="TEX_2:def.5">Def7</a></span>;<br/>
<a NAME="E8:77"/>
<font color="Maroon" title="c3">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="tex_2.html#E40"><span class="lab"><font color="Green" title="E22">Lm3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:77"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:77"><span class="lab"><font color="Green" title="E34">A6</font></span></a></span>;<br/>
<a NAME="E12:77"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:77"><span class="lab"><font color="Green" title="E34">A6</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">then </span><span class="kw">consider </span><font color="Maroon" title="c6">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E14:77"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E15:77"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c3">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="tex_2.html#T26" target="_self" title="TEX_2:th.26">Th32</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E37">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:77_1"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:77_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</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/><a NAME="E3:77_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <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/><span class="kw">hence </span><a NAME="E4:77_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:77"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:77"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E18:77"/><span class="kw">then </span>
( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:77"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E19:77"/><span class="kw">then </span>
( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">G</font> or <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E20:77"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E16:77"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="tsep_1.html#T36" title="TSEP_1:th.36">TSEP_1:36</a></span>;<br/>
<a NAME="E21:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:77_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:77_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:77"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:77"><span class="lab"><font color="Green" title="E36">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E23:77"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="tops_3.html#D2" title="TOPS_3:def.2">TOPS_3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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