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<span class="kw">set </span><font color="Maroon" title="c1">D</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">Y</font> =  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<span class="kw">take </span>
 <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>  &amp; not  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  &amp;  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  &amp; not  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_2_1_1"/>
 <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_2_1_1"/><span class="kw">then </span>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tdlat_3.html#D1" title="TDLAT_3:def.1">TDLAT_3:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">d0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><span class="kw"> such that </span><br/><a NAME="E4:36_2_1_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c3">d0</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="tex_1.html#T23" target="_self" title="TEX_1:th.23">Th23</a></span>;<br/><a NAME="E5:36_2_1_1"/>
<font color="Maroon" title="c3">d0</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="#E4:36_2_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T8" title="ZFMISC_1:th.8">ZFMISC_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:36_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_2_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T8" title="ZFMISC_1:th.8">ZFMISC_1:8</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">hence </span><a NAME="E2:36_2_1"/>
(  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a>  &amp; not  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  &amp;  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  &amp; not  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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/>


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