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<b>let </b><font color="Maroon">K</font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K11" target="_self">VeroneseSpace</a> <font color="Maroon">S</font></span>)</span>, <font color="Maroon">L</font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K11" target="_self">VeroneseSpace</a> <font color="Maroon">S</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D7">PENCIL_1:def 7</a><br/>

<b>assume </b><a NAME="E1:49_1_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2">PENCIL_1:2</a></i>;<br/>
<a NAME="E4:49_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:49_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
not the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K11" target="_self">VeroneseSpace</a> <font color="Maroon">S</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="subset_1.html#D2">SUBSET_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">k</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:49_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K9" target="_self">PairSet</a> <font color="Maroon">t</font>,<font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E13:49_1_1"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">l</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E9:49_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K9" target="_self">PairSet</a> <font color="Maroon">s</font>,<font color="Maroon">l</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E13:49_1_1"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:49_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, , </i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:49_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, , </i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:49_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, , </i>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:49_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">w</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, , </i>;<br/>
<a NAME="E18:49_1_1"/>
( ( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> ) &amp; ( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T10">ZFMISC_1:10</a>, <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E19:49_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E2:49_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T10">ZFMISC_1:10</a>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a>, <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="pencil_4.html#D1" target="_self">Def1</a>, <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a>, <a class="ref" href="zfmisc_1.html#T10">ZFMISC_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:49_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="ref" href="pencil_4.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 ;<br/><a NAME="E2:49_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T10">ZFMISC_1:10</a>, , <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a>, , <a class="ref" href="zfmisc_1.html#T10">ZFMISC_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:49_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">l</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="ref" href="pencil_4.html#T5" target="_self">Th5</a>, , <a class="ref" href="pencil_4.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E24:49_1_1"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T2">PENCIL_1:2</a>, , <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E25:49_1_1"/>
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D7">PENCIL_1:def 7</a>, , , </i>;<br/>


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