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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E10</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E11</font></i>: 
( <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; <font color="Maroon">c<sub>3</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>6</sub></font> &amp; <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>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> <a href="incproj.html#R2">are_mutually_different</a>  )
 ;<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="E4:13"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</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#E1:13"><i><font color="Green">E10</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <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="E3:13_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>9</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>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</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; not <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="projred1.html#T3" target="_self">Th3</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="E5:13_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>9</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#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:13_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <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> &amp; <font color="Maroon">c<sub>12</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="ref" href="projred1.html#T8" target="_self">Th8</a></i>;<br/><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E17</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>
 ;<br/><div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<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="E3:13_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>12</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>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><b>thus </b><a NAME="E4:13_1_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>13</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="txt" href="projred1.html#E4:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green">E19</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> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<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="E3:13_1_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>12</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/><a NAME="E4:13_1_1_2"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E2:13_1_1_2_1"/><b>then </b>
<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#E5:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1_2"><i><font color="Green">E19</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>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="E6:13_1_1_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</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="txt" href="projred1.html#E4:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projred1.html#E3:13_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><b>thus </b><a NAME="E7:13_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E6:13_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:13_1_1"/>
ex <font color="Olive">b<sub>1</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>2</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E2:13_1_1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</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>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:13_1_2"/><i><font color="Green">E18</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>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><b>assume </b><a NAME="E3:13_1_2"/><i><font color="Green">E20</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>14</sub></font>
 ;<br/><a NAME="E4:13_1_2"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>12</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#E3:13_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:13_1_2"/><i><font color="Green">E22</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>6</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projred1.html#E2:13_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>assume </b><a NAME="E6:13_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E7:13_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E2:13_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E4:13_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>hence </b><a NAME="E8:13_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:13_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E2:13_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:13_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/></div><b>end;</b></div><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>, <font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:13_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>2</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="txt" href="projred1.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green">E16</font></i></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>, <font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:13_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>3</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="txt" href="projred1.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green">E16</font></i></a></i>;<br/><b>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>, <font color="Maroon">c<sub>18</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:13_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green">E16</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</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>20</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:13_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E18:13_1"/>
( <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>4</sub></font> &amp; <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>2</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; <font color="Maroon">c<sub>2</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#E2:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="incproj.html#D6">INCPROJ:def 6</a></i>;<br/><a NAME="E19:13_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projred1.html#E9:13_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projred1.html#E11:13_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E13:13_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E15:13_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E17:13_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E20:13_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="incproj.html#R2">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D6">INCPROJ:def 6</a></i>;<br/><b>hence </b><a NAME="E21:13_1"/>
ex <font color="Olive">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> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <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> <a href="incproj.html#R2">are_mutually_different</a>  )
 <b>by </b><i><a class="txt" href="projred1.html#E11:13_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projred1.html#E13:13_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred1.html#E15:13_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E17:13_1"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
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> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <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> <a href="incproj.html#R2">are_mutually_different</a>  )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green">E11</font></i></a></i>;<br/>


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