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<div class="add">

<b>set </b><font color="Maroon">D</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>;<br/>
<b>take </b>
 <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2_1_1"/>
 <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> is <a href="tdlat_3.html#V1">discrete</a>
 ;<br/><a NAME="E2:34_2_1_1"/><b>then </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> #)
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">d0</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:34_2_1_1"/><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:34_2_1_1"/>
( <font color="Maroon">d0</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">d0</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T8">ZFMISC_1:8</a></i>;<br/><b>hence </b><a NAME="E6:34_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2_1"/>
(  <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> is <a href="tdlat_3.html#V2">anti-discrete</a> &amp; not  <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> is <a href="tdlat_3.html#V1">discrete</a> &amp;  <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> is <a href="pre_topc.html#V1">strict</a> &amp; not  <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> is <a href="struct_0.html#V3">empty</a> )
 ;<br/>


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