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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:56"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a>
 ;<br/>

<b>thus </b><a NAME="E2:56"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pencil_1.html#V2" target="_self">strong</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D3" target="_self">PENCIL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>3</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#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E2:56_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:56_1_1_1_2"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:56_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


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

<b>thus </b><a NAME="E3:56"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pencil_1.html#NM1">Block</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D2" target="_self">PENCIL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:56_2"/><i><font color="Green">E38</font></i>: 
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">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:56_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:56_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E4:56_2"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E1:56_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E5:56_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/>


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


</div>
