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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E34</font></i>: 
3 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:13"/>
2 <a href="tarski.html#R1">c=</a> 3
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E3:13"/><b>then </b><i><font color="Green">E35</font></i>: 
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>let </b><font color="Maroon">K</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E4:13"/><i><font color="Green">E37</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> 2
 ;<br/>

<a NAME="E5:13"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/>
<b>let </b><font color="Maroon">S</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E7:13"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:13"/><i><font color="Green">E49</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:13"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">K'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, <a class="ref" href="card_2.html#T79">CARD_2:79</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<b>thus </b><a NAME="E12:13"/>
not <font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</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="E14:13"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E15:13"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:13_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:13_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">l</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E17:13"/><i><font color="Green">E74</font></i>: 
( not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E18:13"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<a NAME="E19:13"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  &amp; not <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:13"/><b>then </b><i><font color="Green">E75</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span>
 
;<br/>
<a NAME="E21:13"/><b>then </b><i><font color="Green">E76</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E22:13"/>
not <font color="Maroon">S</font> is <a href="pencil_1.html#V3" target="_self">void</a>
 <b>by </b><i>, </i>;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span> 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#T37">XBOOLE_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:13_2"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">L</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:13_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> &amp; 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">L</font> )
 ;<br/><b>thus </b><a NAME="E5:13_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:13"/><b>then </b>
<font color="Maroon">X</font> is  not    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E26:13"/>
not <font color="Maroon">S</font> is <a href="pencil_1.html#V4" target="_self">degenerated</a>
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E27:13"/>
<font color="Maroon">S</font> is <a href="pencil_1.html#V7" target="_self">truly-partial</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D8" target="_self">PENCIL_1:def 8</a><br/>

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<a NAME="E1:13_3"/>
for <font color="Olive">l</font> being  <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font>  holds not <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E1:13_3_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  &amp; not <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">L</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:13_3_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/>
<a NAME="E4:13_3_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> 2
 
<b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/>
<b>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="E7:13_3_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">L'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="card_2.html#T79">CARD_2:79</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_3_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:13_3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L'</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E3:13_3_1_1"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:13_3_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:13_3_1"/>
not <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">l</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_3"/>
not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E28:13"/>
<font color="Maroon">S</font> is <a href="pencil_1.html#V5" target="_self">with_non_trivial_blocks</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D6" target="_self">PENCIL_1:def 6</a><br/>

<a NAME="E1:13_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:13_4"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/>
<b>thus </b><a NAME="E4:13_4"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E29:13"/>
<font color="Maroon">S</font> is <a href="pencil_1.html#V6" target="_self">identifying_close_blocks</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font>, <font color="Maroon">l</font> be   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D7" target="_self">PENCIL_1:def 7</a><br/>

<a NAME="E1:13_5"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:13_5"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/>
<b>assume </b><a NAME="E4:13_5"/>
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="E6:13_5"/><i><font color="Green">E85</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/>;<br/>
<a NAME="E7:13_5"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:13_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:13_5_1"/>
<font color="Maroon">x</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>
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:13_5"/><i><font color="Green">E88</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<a NAME="E9:13_5"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/>
<a NAME="E11:13_5"/>
<font color="Maroon">k</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E12:13_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:13_5"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, , <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/>
<a NAME="E14:13_5"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E16:13_5"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/>
<a NAME="E17:13_5"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> 2
 
<b>by </b><i><a class="txt" href="pencil_1.html#E3:13"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">l</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/>
<a NAME="E19:13_5"/>
<font color="Maroon">k</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">l1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E20:13_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">l1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E21:13_5"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="txt" href="pencil_1.html#E1:13"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_1.html#E3:13"><i><font color="Green">E35</font></i></a>, <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/>


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

<b>thus </b><a NAME="E30:13"/>
<font color="Maroon">S</font> is <a href="pencil_1.html#V8" target="_self">without_isolated_points</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D9" target="_self">PENCIL_1:def 9</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_6_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> ) or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_6_1_1"><b>suppose </b></a><a NAME="E1:13_6_1_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_6_1_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:13_6_1_1"/><i><font color="Green">E100</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/><a NAME="E5:13_6_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:13_6_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:13_6_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:13_6_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">el</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E7:13_6_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">el</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E8:13_6_1_1"/><b>then </b>
<font color="Maroon">el</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:13_6_1_1"/><b>then </b>
( <font color="Maroon">el</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  &amp; not <font color="Maroon">el</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">el</font> = <font color="Maroon">el</font> as   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>take </b>
<font color="Maroon">el</font>
;<br/><b>thus </b><a NAME="E11:13_6_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">el</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_6_1_2"><b>suppose </b></a><a NAME="E1:13_6_1_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_6_1_2"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:13_6_1_2"/><i><font color="Green">E104</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/><a NAME="E5:13_6_1_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_6_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:13_6_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:13_6_1_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:13_6_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">el</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E7:13_6_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">el</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E8:13_6_1_2"/><b>then </b>
<font color="Maroon">el</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:13_6_1_2"/><b>then </b>
( <font color="Maroon">el</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  &amp; not <font color="Maroon">el</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">K</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">el</font> = <font color="Maroon">el</font> as   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>take </b>
<font color="Maroon">el</font>
;<br/><b>thus </b><a NAME="E11:13_6_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">el</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
