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

<b>let </b><font color="Maroon">G</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">E36</font></i>: 
<font color="Maroon">G</font> is <a href="projpl_1.html#V1" target="_self">configuration</a>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>  ex <font color="Olive">M</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <font color="Olive">p</font>,<font color="Olive">q</font> <a href="incproj.html#R3">on</a> <font color="Olive">M</font>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  ex <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <font color="Olive">a</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Olive">P</font>,<font color="Olive">Q</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>,<font color="Olive">d</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">P</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>, <font color="Maroon">d</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:31_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:31_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/><b>thus </b><a NAME="E4:31_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</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">a</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font> or <font color="Maroon">b</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font> or <font color="Maroon">c</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font> )
 </span><b>by </b><i>, </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">E44</font></i>: 
<font color="Maroon">a</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">B</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:31_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R3">on</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:31_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="incproj.html#R3">on</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:31_1_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="incproj.html#R3">on</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:31_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, <a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E2:31"><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">E48</font></i>: 
<font color="Maroon">a</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font>
 <b>by </b><i/>;<br/><a NAME="E10:31_1_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, <a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, <a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E37</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E12:31_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E14:31_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">q</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E16:31_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E17:31_1_1_1_1_1"/>
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E4:31"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E18:31_1_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</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">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#T3" target="_self">Th3</a>, <a class="ref" href="projpl_1.html#T4" target="_self">Th4</a>, <a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E4:31"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E20:31_1_1_1_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_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">E54</font></i>: 
<font color="Maroon">b</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">B</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:31_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="incproj.html#R3">on</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:31_1_1_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R3">on</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:31_1_1_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="incproj.html#R3">on</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:31_1_1_1_1_2"/>
( <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, <a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E2:31"><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">E58</font></i>: 
<font color="Maroon">b</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font>
 <b>by </b><i/>;<br/><a NAME="E10:31_1_1_1_1_2"/>
<font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E11:31_1_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, <a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E37</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E13:31_1_1_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">p</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E15:31_1_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">q</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E17:31_1_1_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">r</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E18:31_1_1_1_1_2"/>
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E4:31"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E19:31_1_1_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</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">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#T3" target="_self">Th3</a>, <a class="ref" href="projpl_1.html#T4" target="_self">Th4</a>, <a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E4:31"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E21:31_1_1_1_1_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_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_3"><b>case </b></a><a NAME="E1:31_1_1_1_1_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">B</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:31_1_1_1_1_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="incproj.html#R3">on</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:31_1_1_1_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="incproj.html#R3">on</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:31_1_1_1_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="incproj.html#R3">on</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:31_1_1_1_1_3"/>
( <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, <a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E2:31"><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">E90</font></i>: 
<font color="Maroon">c</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font>
 <b>by </b><i/>;<br/><a NAME="E10:31_1_1_1_1_3"/>
<font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E11:31_1_1_1_1_3"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, <a class="txt" href="projpl_1.html#E1:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projpl_1.html#E2:31"><i><font color="Green">E37</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E13:31_1_1_1_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">p</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E15:31_1_1_1_1_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">q</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E17:31_1_1_1_1_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">r</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E18:31_1_1_1_1_3"/>
( <font color="Maroon">p</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E4:31"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E19:31_1_1_1_1_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</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">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i>, <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, <a class="ref" href="projpl_1.html#T3" target="_self">Th3</a>, <a class="ref" href="projpl_1.html#T4" target="_self">Th4</a>, <a class="txt" href="projpl_1.html#E3:31"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projpl_1.html#E4:31"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E21:31_1_1_1_1_3"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1"><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">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> )
 ;<br/>


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


</div>
