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

<b>let </b><font color="Maroon" title="c10">CLSP</font> be  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a> <a href="collsp.html#NM2" title="COLLSP:NM.2">CollSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><br/> for <font color="Olive" title="b3">P</font> being   <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> holds <br/> <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font></span><br/><b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</font>  st <font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> holds <br/> <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c13">P</font> be    <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font> implies  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">P</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">P</font></span><br/>

<b>consider </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><b> such that </b><br/><a NAME="E5:36"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b</font>
 <b>and </b><br/><a NAME="E6:36"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="collsp.html#D7" target="_self" title="COLLSP:def.7">Def7</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c11">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c12">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="collsp.html#E2:36"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="collsp.html#E3:36"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="collsp.html#E6:36"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="collsp.html#T17" target="_self" title="COLLSP:th.17">Th17</a></i>;<br/>
<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font> )</span><br/>

<b>assume </b><a NAME="E8:36"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c17">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><b> such that </b><br/><a NAME="E10:36"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 <b>and </b><br/><a NAME="E11:36"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c17">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>
<a NAME="E12:36"/>
<font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c17">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="collsp.html#E1:36"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="collsp.html#E7:36"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="collsp.html#E11:36"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="collsp.html#T14" target="_self" title="COLLSP:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E13:36"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 <b>by </b><i><a class="txt" href="collsp.html#E6:36"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="collsp.html#E10:36"><i><font color="Green" title="E34">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
