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<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<a NAME="E1:65"/><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_3.html#D8" target="_self">Def8</a></i>;<br/>
<b>thus </b><a NAME="E2:65"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E3:65_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; ( ( <font color="Maroon">c<sub>5</sub></font> is <a href="tex_2.html#V1">proper</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> ) or  not <font color="Maroon">c<sub>5</sub></font> is <a href="tex_2.html#V1">proper</a> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:65"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topgen_3.html#E1:65_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topgen_3.html#E60"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:65_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:65"><i><font color="Green">E48</font></i></a>, <a class="ref" href="topgen_3.html#T43" target="_self">Th43</a>, <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<a NAME="E5:65_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:65_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:65"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E4:65"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:65"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topgen_3.html#E1:65"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E7:65"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E8:65"/><b>then </b><i><font color="Green">E50</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:65"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<a NAME="E10:65"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:65"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E11:65"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="tex_2.html#V1">proper</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:65"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<a NAME="E12:65"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:65"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topgen_3.html#E60"><i><font color="Green">Lemma46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:65"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:65"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topgen_3.html#E8:65"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topgen_3.html#E11:65"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topgen_3.html#E10:65"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topgen_3.html#T43" target="_self">Th43</a></i>;<br/>


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