<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">L</font> be   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K11" title="PENCIL_4:func.11">VeroneseSpace</a> <font color="Maroon" title="c1">S</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D7" title="PENCIL_1:def.7">PENCIL_1:def 7</a><br/>

<b>assume </b><a NAME="E1:49_1_1"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">L</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:49_1_1"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></i>;<br/>
<a NAME="E4:49_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E3:49_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:49_1_1"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
not the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K11" title="PENCIL_4:func.11">VeroneseSpace</a> <font color="Maroon" title="c1">S</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="subset_1.html#D2" title="SUBSET_1:def.2">SUBSET_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c7">k</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:49_1_1"/><i><font color="Green" title="E38">A4</font></i>: 
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> <font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c7">k</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D10" target="_self" title="PENCIL_4:def.10">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">l</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E9:49_1_1"/><i><font color="Green" title="E39">A5</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> <font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">l</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D10" target="_self" title="PENCIL_4:def.10">Def10</a>, <a class="txt" href="pencil_4.html#E5:49_1_1"><i><font color="Green" title="E37">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:49_1_1"/><i><font color="Green" title="E40">A6</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E4:49_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="pencil_4.html#E7:49_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="pencil_4.html#D9" target="_self" title="PENCIL_4:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:49_1_1"/><i><font color="Green" title="E41">A7</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E4:49_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="pencil_4.html#E7:49_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="pencil_4.html#D9" target="_self" title="PENCIL_4:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:49_1_1"/><i><font color="Green" title="E42">A8</font></i>: 
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c12">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E4:49_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="pencil_4.html#E9:49_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="pencil_4.html#D9" target="_self" title="PENCIL_4:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:49_1_1"/><i><font color="Green" title="E43">A9</font></i>: 
( <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c13">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E4:49_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="pencil_4.html#E9:49_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="pencil_4.html#D9" target="_self" title="PENCIL_4:def.9">Def9</a></i>;<br/>
<a NAME="E18:49_1_1"/><i><font color="Green" title="E44">A10</font></i>: 
( ( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font> ) &amp; ( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">w</font> ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a>, <a class="txt" href="pencil_4.html#E11:49_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="pencil_4.html#E13:49_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="pencil_4.html#E15:49_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="pencil_4.html#E17:49_1_1"><i><font color="Green" title="E43">A9</font></i></a></i>;<br/>
<div><i><font color="Green" title="E45">A11</font></i>: <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" title="E46">A12</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">z</font>
 ;<br/><a NAME="E2:49_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:49_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="pencil_4.html#E11:49_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="pencil_4.html#E15:49_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="pencil_4.html#E17:49_1_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="pencil_4.html#E18:49_1_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:49_1_1_1"><i><font color="Green" title="E46">A12</font></i></a>, <a class="txt" href="pencil_4.html#E11:49_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="pencil_4.html#E15:49_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a></i>;<br/></div><b>end;</b></div>
<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" title="E46">A13</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">w</font>
 ;<br/><a NAME="E2:49_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:49_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="pencil_4.html#E13:49_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="pencil_4.html#E15:49_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="pencil_4.html#E17:49_1_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="pencil_4.html#E18:49_1_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:49_1_1_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="pencil_4.html#E13:49_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="pencil_4.html#E17:49_1_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:49_1_1"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">l</font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E11:49_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="pencil_4.html#E13:49_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="pencil_4.html#E15:49_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="pencil_4.html#E17:49_1_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="pencil_4.html#E19:49_1_1"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:49_1_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">l</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a>, <a class="txt" href="pencil_4.html#E3:49_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="pencil_4.html#E11:49_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="pencil_4.html#E13:49_1_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:49_1_1"/>
<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:49_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="pencil_4.html#E7:49_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="pencil_4.html#E9:49_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="pencil_4.html#E15:49_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="pencil_4.html#E17:49_1_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="pencil_4.html#E18:49_1_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="pencil_1.html#D7" title="PENCIL_1:def.7">PENCIL_1:def 7</a></i>;<br/>


</div>
