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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="incproj.html#V9">Fanoian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>3</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="E2:15"/><i><font color="Green">E12</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="incproj.html#D11">INCPROJ:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="E4:15"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>4</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>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D12">INCPROJ:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</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="E6:15"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</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#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E8:15"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</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>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="projred1.html#T8" target="_self">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</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="E10:15"/><i><font color="Green">E16</font></i>: 
( <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>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</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>10</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="E12:15"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</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>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="E14:15"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>2</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="E16:15"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</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/>
<a NAME="E17:15"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:15"/><b>then </b><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E19:15"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E20:15"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E21:15"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:15_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:15_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:15"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E2:15_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:15_3"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:15"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E24:15"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:15_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:15_4"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <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">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="E26:15"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E23:15"><i><font color="Green">E26</font></i></a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/>
<a NAME="E27:15"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green">E25</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E28:15"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</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="E30:15"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E27:15"><i><font color="Green">E28</font></i></a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</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="E32:15"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E33:15"/><i><font color="Green">E32</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E23:15"><i><font color="Green">E26</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E34:15"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E28:15"><i><font color="Green">E29</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E35:15"/><b>then </b><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E36:15"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E28:15"><i><font color="Green">E29</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E37:15"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green">E30</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E38:15"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>
<a NAME="E39:15"/><b>then </b><i><font color="Green">E38</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E18:15"><i><font color="Green">E21</font></i></a>, <a class="txt" href="projred1.html#E19:15"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projred1.html#E21:15"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E33:15"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E35:15"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projred1.html#E36:15"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projred1.html#E37:15"><i><font color="Green">E36</font></i></a>, <a class="ref" href="incproj.html#D18">INCPROJ:def 18</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="E41:15"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</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/>
<a NAME="E42:15"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E23:15"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E43:15"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E34:15"><i><font color="Green">E33</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E44:15"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E45:15"/><i><font color="Green">E43</font></i>: 
not <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="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E43:15"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E46:15"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E47:15"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E48:15"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E12:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E42:15"><i><font color="Green">E40</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E49:15"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E43:15"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E50:15"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="incproj.html#R2">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E39:15"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E45:15"><i><font color="Green">E43</font></i></a>, <a class="txt" href="projred1.html#E46:15"><i><font color="Green">E44</font></i></a>, <a class="txt" href="projred1.html#E47:15"><i><font color="Green">E45</font></i></a>, <a class="txt" href="projred1.html#E48:15"><i><font color="Green">E46</font></i></a>, <a class="ref" href="incproj.html#D6">INCPROJ:def 6</a></i>;<br/>
<a NAME="E51:15"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E17:15"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green">E30</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E52:15"/><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>15</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E43:15"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</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="E54:15"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E44:15"><i><font color="Green">E42</font></i></a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/>
<a NAME="E55:15"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projred1.html#E21:15"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E51:15"><i><font color="Green">E49</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green">E50</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E56:15"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E14:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E49:15"><i><font color="Green">E47</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green">E50</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E57:15"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R2">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E39:15"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green">E50</font></i></a>, <a class="txt" href="projred1.html#E55:15"><i><font color="Green">E51</font></i></a>, <a class="ref" href="incproj.html#D6">INCPROJ:def 6</a></i>;<br/>
<b>hence </b><a NAME="E58:15"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font>, <font color="Olive">b<sub>10</sub></font>, <font color="Olive">b<sub>11</sub></font>, <font color="Olive">b<sub>12</sub></font>, <font color="Olive">b<sub>13</sub></font>, <font color="Olive">b<sub>14</sub></font>, <font color="Olive">b<sub>15</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>16</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>12</sub></font> &amp; not <font color="Olive">b<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>12</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>11</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>11</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>13</sub></font> &amp; not <font color="Olive">b<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>13</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>14</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>14</sub></font> &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>12</sub></font> &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>11</sub></font> &amp; <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>13</sub></font> &amp; <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>14</sub></font> &amp; <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="incproj.html#R4">on</a> <font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>10</sub></font> &amp; not <font color="Olive">b<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>15</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>15</sub></font> &amp; <font color="Olive">b<sub>10</sub></font>,<font color="Olive">b<sub>15</sub></font>,<font color="Olive">b<sub>13</sub></font>,<font color="Olive">b<sub>14</sub></font> <a href="incproj.html#R2">are_mutually_different</a>  &amp; <font color="Olive">b<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>8</sub></font> &amp; <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="incproj.html#R2">are_mutually_different</a>  )
 <b>by </b><i><a class="txt" href="projred1.html#E18:15"><i><font color="Green">E21</font></i></a>, <a class="txt" href="projred1.html#E19:15"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projred1.html#E21:15"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E33:15"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E35:15"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projred1.html#E36:15"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projred1.html#E37:15"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projred1.html#E38:15"><i><font color="Green">E37</font></i></a>, <a class="txt" href="projred1.html#E39:15"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E50:15"><i><font color="Green">E48</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green">E50</font></i></a></i>;<br/>


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