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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="incproj.html#V6">2-dimensional</a> <a href="incproj.html#V10">Desarguesian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

















<b>assume </b><a NAME="E1:27"/><i><font color="Green">E20</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:27"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:27"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:27"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
( not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="projred2.html#E1:27"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred2.html#E3:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="projred2.html#E5:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred2.html#E7:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projred2.html#E9:27"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred2.html#E11:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred2.html#E13:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred2.html#E15:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E3:27_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="projred2.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:27_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred2.html#E1:27"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred2.html#E3:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="projred2.html#E5:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred2.html#E7:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projred2.html#E9:27"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred2.html#E11:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred2.html#E13:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred2.html#E15:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred2.html#E1:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="projred2.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E5:27_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E15:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred2.html#E3:27_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:27"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="projred2.html#E1:27"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
