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<b>let </b><font color="Maroon">IPP</font> be  <a href="incproj.html#V10">Desarguesian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">b1</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">a1</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">b2</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">a2</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">b3</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">a3</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">r</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">s</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">t</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">C1</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">C2</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">C3</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">A1</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">A2</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">A3</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">B1</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">B2</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>, <font color="Maroon">B3</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">b1</font>,<font color="Maroon">a1</font> <a href="incproj.html#R4">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="incproj.html#R4">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="incproj.html#R4">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">r</font>,<font color="Maroon">a1</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">s</font>,<font color="Maroon">a1</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">t</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">r</font>,<font color="Maroon">b3</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">s</font>,<font color="Maroon">b2</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">C1</font>,<font color="Maroon">C2</font>,<font color="Maroon">C3</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E4:12"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E5:12"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">C3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C1</font> )
 
<b>by </b><i>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/>
<a NAME="E6:12"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B3</font> )
 
<b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E2:12_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_1_1_1"/>
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_1_1_1"/>
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:12_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>consider </b><font color="Maroon">O</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E9:12_1_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">O</font> = <font color="Maroon">O</font>;<br/><b>thus </b><a NAME="E10:12_1_1_1"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E2:12_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_1_1_2"/>
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>take </b><font color="Maroon">C2</font> = <font color="Maroon">C2</font>;<br/><b>thus </b><a NAME="E7:12_1_1_2"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , , , , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_1_1"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><b>now </b></a><div class="add"><div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E2:12_1_2_1_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A3</font>
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_1_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_1_2_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_1_2_1_1"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_1_2_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">s</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>take </b><font color="Maroon">B2</font> = <font color="Maroon">B2</font>;<br/><b>thus </b><a NAME="E7:12_1_2_1_1"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_2_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E2:12_1_2_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_1_2_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_1_2_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_1_2_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_1_2_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:12_1_2_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">t</font>,<font color="Maroon">s</font>,<font color="Maroon">r</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A3</font>
 <b>by </b><i>, , , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>take </b><font color="Maroon">A3</font> = <font color="Maroon">A3</font>;<br/><b>thus </b><a NAME="E8:12_1_2_1_2"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">A3</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:12_1_2_1"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:12_1_2"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_1"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> )
 ;<br/><div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 ;<br/><a NAME="E2:12_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:12_2_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>consider </b><font color="Maroon">O</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E9:12_2_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">O</font> = <font color="Maroon">O</font>;<br/><b>thus </b><a NAME="E10:12_2_1"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_2_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_2_2_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_2_2_1_1"/>
( <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> )
 
<b>by </b><i>, , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:12_2_2_1_1"/>
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E3:12_2_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_2_2_1"/>
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_2_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_2_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>consider </b><font color="Maroon">O</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E8:12_2_2_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">O</font> = <font color="Maroon">O</font>;<br/><b>thus </b><a NAME="E9:12_2_2_1"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_2_2"/>
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_2_2_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_2_2_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_2_2_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">B3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_2_2_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_2_2_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:12_2_2_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">t</font>,<font color="Maroon">s</font>,<font color="Maroon">r</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>take </b><font color="Maroon">B1</font> = <font color="Maroon">B1</font>;<br/><b>thus </b><a NAME="E8:12_2_2_2"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_2_2"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_2"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E108</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> )
 ;<br/><div><i><font color="Green">E110</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/><div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_3_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">A3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_1_1_1"/>
( <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> )
 
<b>by </b><i>, , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:12_3_1_1_1"/>
<font color="Maroon">A3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B3</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E3:12_3_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_1_1_2"/>
( <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> )
 
<b>by </b><i>, , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:12_3_1_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:12_3_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_3_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">s</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>take </b><font color="Maroon">B2</font> = <font color="Maroon">B2</font>;<br/><b>thus </b><a NAME="E6:12_3_1_1"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_3_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_3_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_1_2_1"/>
( <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, , , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:12_3_1_2_1"/>
<font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:12_3_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">B3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_3_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_3_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>consider </b><font color="Maroon">O</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E8:12_3_1_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">O</font> = <font color="Maroon">O</font>;<br/><b>thus </b><a NAME="E9:12_3_1_2"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_3_1"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font>
 ;<br/><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_2_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_3_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_3_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_2_1_1"/>
( <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, , , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:12_3_2_1_1"/>
<font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:12_3_2_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_3_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_3_2_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>consider </b><font color="Maroon">O</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="E8:12_3_2_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">O</font> = <font color="Maroon">O</font>;<br/><b>thus </b><a NAME="E9:12_3_2_1"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_2_2"/>
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_3_2_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E3:12_3_2_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E4:12_3_2_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">B3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C1</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_3_2_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:12_3_2_2"/>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:12_3_2_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">s</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, , , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>take </b><font color="Maroon">B2</font> = <font color="Maroon">B2</font>;<br/><b>thus </b><a NAME="E8:12_3_2_2"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_3_2"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_3"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E136</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_4"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 ;<br/><div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_4_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_4_1"/><b>then </b>
( <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E3:12_4_1"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B1</font>
 <b>by </b><i/>;<br/><a NAME="E4:12_4_1"/>
( <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> )
 <b>by </b><i>, , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E5:12_4_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">A3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B3</font>
 <b>by </b><i/>;<br/><a NAME="E6:12_4_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:12_4_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">s</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>take </b><font color="Maroon">B2</font> = <font color="Maroon">B2</font>;<br/><b>thus </b><a NAME="E8:12_4_1"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_4_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font>
 ;<br/><a NAME="E2:12_4_2"/>
( <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A3</font> )
 <b>by </b><i>, , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E3:12_4_2"/><b>then </b>
<font color="Maroon">A3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B3</font>
 <b>by </b><i/>;<br/><a NAME="E4:12_4_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:12_4_2"/>
( <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">a3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, , , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E6:12_4_2"/><b>then </b>
<font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <b>by </b><i/>;<br/><a NAME="E7:12_4_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>consider </b><font color="Maroon">O</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E9:12_4_2"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">O</font> = <font color="Maroon">O</font>;<br/><b>thus </b><a NAME="E10:12_4_2"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_4"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_5"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font> )
 ;<br/><a NAME="E2:12_5"/><b>then </b><i><font color="Green">E148</font></i>: 
( <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">a1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b1</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E3:12_5"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">a2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">b2</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">b3</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C3</font> &amp; <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, , <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E4:12_5"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:12_5"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:12_5"/>
<font color="Maroon">b3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:12_5"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>consider </b><font color="Maroon">O</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="E9:12_5"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">t</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> &amp; <font color="Maroon">s</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon">O</font> = <font color="Maroon">O</font>;<br/><b>thus </b><a NAME="E10:12_5"/>
<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:12"/>
 ex <font color="Olive">O</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font> st <font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">t</font> <a href="incproj.html#R4">on</a> <font color="Olive">O</font>
 <b>by </b><i>, , , , , <a class="ref" href="incproj.html#D19">INCPROJ:def 19</a></i>;<br/>


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