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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">A</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a> &amp; not <font color="Olive" title="b1">A</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> holds <br/> not <font color="Olive" title="b1">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a> &amp; not <font color="Maroon" title="c2">A</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> implies  not <font color="Maroon" title="c2">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a> )</span><br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> or  not <font color="Maroon" title="c2">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a> )</span><br/>

<b>assume </b><a NAME="E2:28"/><i><font color="Green" title="E20">A2</font></i>: 
not <font color="Maroon" title="c2">A</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</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="c2">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a></span><br/>

<b>consider </b><font color="Maroon" title="c3">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">s0</font> = <font color="Maroon" title="c3">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="tsp_1.html#E4:28"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>
<a NAME="E6:28"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">s0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_1.html#E2:28"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:28"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E9:28"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">s0</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T41" title="ZFMISC_1:th.41">ZFMISC_1:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c5">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="tsp_1.html#E4:28"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="tsp_1.html#E8:28"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<b>assume </b><a NAME="E11:28"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1"/>
(  ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; not <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> ) or  ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) )
 </span><b>by </b><i><a class="txt" href="tsp_1.html#E8:28"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="tsp_1.html#E9:28"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="tsp_1.html#E11:28"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="tsp_1.html#D9" target="_self" title="TSP_1:def.9">Def9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1"/>
 ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; not <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:28_1_1_1"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c8">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:28_1_1_1"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 <b>and </b><br/><a NAME="E5:28_1_1_1"/><i><font color="Green" title="E27">A9</font></i>: 
not <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 ;<br/><a NAME="E6:28_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:28"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="tsp_1.html#E4:28"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="tsp_1.html#E3:28_1_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="tsp_1.html#E4:28_1_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="tex_4.html#D2" title="TEX_4:def.2">TEX_4:def 2</a></i>;<br/><b>hence </b><a NAME="E7:28_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_1.html#E8:28"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="tsp_1.html#E5:28_1_1_1"><i><font color="Green" title="E27">A9</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">F</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:28_1_1_2"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c8">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:28_1_1_2"/><i><font color="Green" title="E26">A11</font></i>: 
not <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font>
 <b>and </b><br/><a NAME="E5:28_1_1_2"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font>
 ;<br/><a NAME="E6:28_1_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:28"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="tsp_1.html#E8:28"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="tsp_1.html#E3:28_1_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="tsp_1.html#E5:28_1_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="tex_4.html#D2" title="TEX_4:def.2">TEX_4:def 2</a></i>;<br/><b>hence </b><a NAME="E7:28_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:28"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="tsp_1.html#E4:28_1_1_2"><i><font color="Green" title="E26">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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:28"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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