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

<b>let </b><font color="Maroon">G</font> be   <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R1">are_mutually_different</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:36_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>thus </b><a NAME="E4:36_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:36_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>thus </b><a NAME="E4:36_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_3"><b>case </b></a><a NAME="E1:36_1_1_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:36_1_1_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>thus </b><a NAME="E4:36_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
contradiction
 ;<br/>


</div>
