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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:59"/><i><font color="Green">E43</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#K6" target="_self">-PointClTop</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#D7" target="_self">Def7</a></i>;<br/>
<b>assume </b><a NAME="E2:59"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T62">XBOOLE_1:62</a></i>;<br/>
<a NAME="E4:59"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="topgen_3.html#E2:59"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E6:59"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="tex_2.html#V1">proper</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="topgen_3.html#E2:59"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> is  <a href="pre_topc.html#V3">open</a> <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#K6" target="_self">-PointClTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>hence </b><a NAME="E2:59_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>assume </b><a NAME="E3:59_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:59_1"/><b>then </b>
(  not <font color="Maroon">c<sub>3</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>3</sub></font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>8</sub></font> is  non <a href="tex_2.html#V1">proper</a> <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#K6" target="_self">-PointClTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:59_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topgen_3.html#T39" target="_self">Th39</a></i>;<br/><b>hence </b><a NAME="E5:59_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E9:59"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E10:59"/><b>then </b><i><font color="Green">E46</font></i>: 
( not <font color="Maroon">c<sub>3</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>4</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E6:59"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:59"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> is  <a href="tex_2.html#V1">proper</a> <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#K6" target="_self">-PointClTop</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<b>hence </b><a NAME="E12:59"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> is  <a href="pre_topc.html#V3">open</a> <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#K6" target="_self">-PointClTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E10:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topgen_3.html#T39" target="_self">Th39</a></i>;<br/>


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