<?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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">G</font> is   <a href="projpl_1.html#NM1">IncProjectivePlane</a>
 ;<br/><b>hence </b><a NAME="E2:32_1"/>
<font color="Maroon">G</font> is <a href="projpl_1.html#V1" target="_self">configuration</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:32_1"/>
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>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a>, </i>;<br/><b>thus </b><a NAME="E4:32_1"/>
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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><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>let </b><font color="Maroon">Q</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><b> such that </b><br/><a NAME="E2:32_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a>, <a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E3:32_1_1"/>
<font color="Maroon">a</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:32_1"/>
 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> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a>, </i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:32_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">G</font> is <a href="projpl_1.html#V1" target="_self">configuration</a>
 <b>and </b><br/><a NAME="E2:32_2"/><i><font color="Green">E41</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="E3:32_2"/><i><font color="Green">E42</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="E4:32_2"/><i><font color="Green">E43</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/><a NAME="E5:32_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 <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">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/><a NAME="E6:32_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">p</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">P</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E7:32_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>  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="Olive">P</font> )
 <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a>, <a class="ref" href="projpl_1.html#D3" target="_self">Def3</a>, <a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E8:32_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font><br/> for <font color="Olive">M</font>, <font color="Olive">N</font>, <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>  st <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">p</font> <a href="incproj.html#R4">on</a> <font color="Olive">M</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">p</font> <a href="incproj.html#R4">on</a> <font color="Olive">N</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="incproj.html#R3">on</a> <font color="Olive">P</font> &amp; <font color="Olive">b</font>,<font color="Olive">d</font> <a href="incproj.html#R3">on</a> <font color="Olive">Q</font> &amp; <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">P</font> &amp; <font color="Olive">p</font> <a href="projpl_1.html#NR1" target="_self">|'</a> <font color="Olive">Q</font> &amp; <font color="Olive">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">N</font> holds <br/> ex <font color="Olive">q</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font> st <font color="Olive">q</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Olive">P</font>,<font color="Olive">Q</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as   <a href="incproj.html#NM2">IncProjSp</a> <b>by </b><i><a class="ref" href="projpl_1.html#D1" target="_self">Def1</a>, <a class="ref" href="projpl_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="projpl_1.html#D4" target="_self">Def4</a>, </i>;<br/><a NAME="E10:32_2"/>
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 <br/>( <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">Q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><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>, <font color="Maroon">Q</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><b> such that </b><br/><a NAME="E2:32_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E3:32_2_1"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="projpl_1.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:32_2"/>
<font color="Maroon">G</font> is   <a href="projpl_1.html#NM1">IncProjectivePlane</a>
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/></div>
<b>end;</b></div>


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