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<b>let </b><font color="Maroon">IPP</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">IPP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E33</font></i>: 
 ex <font color="Olive">o</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Olive">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="incproj.html#R2">are_mutually_different</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">o</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">y</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E5:13_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">y</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon">w</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E7:13_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">u</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 ;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">u</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">o</font>;<br/><b>take </b><font color="Maroon">D</font> = <font color="Maroon">D</font>;<br/><b>thus </b><a NAME="E4:13_1_1_1"/>
( <font color="Maroon">v</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">u</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">v</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font> )
 ;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E3:13_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">u</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><a NAME="E4:13_1_1_2"/>
not <font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1_2_1"/>
<font color="Maroon">x</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:13_1_1_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_2_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">v</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E6:13_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">v</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">v</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>take </b><font color="Maroon">D</font> = <font color="Maroon">D</font>;<br/><b>thus </b><a NAME="E7:13_1_1_2"/>
( <font color="Maroon">v</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">u</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">v</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:13_1_1"/>
 ex <font color="Olive">v</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ex <font color="Olive">D</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Olive">v</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">D</font> &amp; <font color="Maroon">u</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">D</font> &amp; <font color="Olive">v</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">D</font> )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u1</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">u2</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">v1</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">v2</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">D1</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">D2</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:13_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">u1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">v1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">u1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">v1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">v2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">u2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">v2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D2</font> )
 ;<br/><b>assume </b><a NAME="E3:13_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u2</font>
 ;<br/><a NAME="E4:13_1_2"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:13_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>assume </b><a NAME="E6:13_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><a NAME="E7:13_1_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>hence </b><a NAME="E8:13_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">p</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">D1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E11:13_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D1</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">D2</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E13:13_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D2</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">D3</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E15:13_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D3</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D3</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D3</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">D4</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E17:13_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D4</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D4</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D4</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E18:13_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D6">INCPROJ:def 6</a></i>;<br/><a NAME="E19:13_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , , , , </i>;<br/><a NAME="E20:13_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="incproj.html#R2">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D6">INCPROJ:def 6</a></i>;<br/><b>hence </b><a NAME="E21:13_1"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">r</font>,<font color="Olive">s</font> <a href="incproj.html#R2">are_mutually_different</a>  )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Olive">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">r</font>,<font color="Olive">s</font> <a href="incproj.html#R2">are_mutually_different</a>  )
 <b>by </b><i/>;<br/>


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