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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="incsp_1.html#L1">IncProjStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="projpl_1.html#V1" target="_self">configuration</a>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>3</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="incproj.html#R3">on</a> <font color="Olive">b<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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>ex <font color="Olive">b<sub>3</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E31</font></i>: 
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 <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="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font>, <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="E2:31_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E4:31"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E3:31_1"/><i><font color="Green">E33</font></i>: 
<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="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="projpl_1.html#D6" target="_self">Def6</a></i>;<br/><b>thus </b><a NAME="E4:31_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> 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>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &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> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="projpl_1.html#T5" target="_self">Th5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><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="E3:31_1_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</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="E5:31_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></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="E7:31_1_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E8:31_1_1_1_1_1"/>
( <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; <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>3</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="projpl_1.html#E3:31_1_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/><a NAME="E9:31_1_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E10:31_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<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="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projpl_1.html#E3:31_1_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="projpl_1.html#T20" target="_self">Th20</a></i>;<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="E12:31_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</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="E14:31_1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></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="E16:31_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E17:31_1_1_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E12:31_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E14:31_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E16:31_1_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E18:31_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E19:31_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projpl_1.html#E1:31_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projpl_1.html#E9:31_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projpl_1.html#E10:31_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projpl_1.html#E12:31_1_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E14:31_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E16:31_1_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projpl_1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E20:31_1_1_1_1_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> 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>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &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> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E18:31_1_1_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><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="E3:31_1_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</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="E5:31_1_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></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="E7:31_1_1_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E8:31_1_1_1_1_2"/>
( <font color="Maroon">c<sub>4</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> <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>9</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/><a NAME="E9:31_1_1_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E10:31_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="projpl_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E11:31_1_1_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<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="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="projpl_1.html#T20" target="_self">Th20</a></i>;<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="E13:31_1_1_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</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="E15:31_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></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="E17:31_1_1_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E18:31_1_1_1_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E13:31_1_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E19:31_1_1_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E20:31_1_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projpl_1.html#E1:31_1_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projpl_1.html#E9:31_1_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projpl_1.html#E11:31_1_1_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projpl_1.html#E13:31_1_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projpl_1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E21:31_1_1_1_1_2"/>
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> 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>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &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> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E19:31_1_1_1_1_2"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_3"><b>case </b></a><a NAME="E1:31_1_1_1_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><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="E3:31_1_1_1_1_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</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="E5:31_1_1_1_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></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="E7:31_1_1_1_1_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="incproj.html#R3">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E8:31_1_1_1_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</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="txt" href="projpl_1.html#E3:31_1_1_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/><a NAME="E9:31_1_1_1_1_3"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E10:31_1_1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="projpl_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E11:31_1_1_1_1_3"/><b>then </b><i><font color="Green">E39</font></i>: 
<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="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1_1_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="projpl_1.html#T20" target="_self">Th20</a></i>;<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="E13:31_1_1_1_1_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</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="E15:31_1_1_1_1_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></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="E17:31_1_1_1_1_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E18:31_1_1_1_1_3"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E13:31_1_1_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E19:31_1_1_1_1_3"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E20:31_1_1_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projpl_1.html#E1:31_1_1_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projpl_1.html#E9:31_1_1_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projpl_1.html#E11:31_1_1_1_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projpl_1.html#E13:31_1_1_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projpl_1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E21:31_1_1_1_1_3"/>
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> 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>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &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> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E19:31_1_1_1_1_3"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_1_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> 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>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &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> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>


</div>
