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<b>let </b><font color="Maroon" title="c1">G</font> be    <a href="incsp_1.html#L1" title="INCSP_1:struct.1">IncProjStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="projpl_1.html#V1" title="PROJPL_1:attr.1">configuration</a>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b3">M</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font> st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b3">M</font>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b3">a</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b3">a</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green" title="E31">A4</font></i>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="projpl_1.html#R5" title="PROJPL_1:pred.5">is_a_quadrangle</a> 
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font>;<br/><b>consider </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:31_1"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="projpl_1.html#R5" title="PROJPL_1:pred.5">is_a_quadrangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E4:31"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/><a NAME="E3:31_1"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="projpl_1.html#NR6" title="PROJPL_1:NR.6">is_a_triangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="projpl_1.html#D6" target="_self" title="PROJPL_1:def.6">Def6</a></i>;<br/><b>thus </b><a NAME="E4:31_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">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" title="c3">a</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c4">b</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c5">c</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c2">P</font> )
 </span><b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="projpl_1.html#T5" target="_self" title="PROJPL_1:th.5">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" title="E34">A7</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c2">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:31_1_1_1_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">C</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:31_1_1_1_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">D</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:31_1_1_1_1_1"/><i><font color="Green" title="E37">A10</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><a NAME="E8:31_1_1_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c3">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c3">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">D</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1_1_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/><a NAME="E9:31_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E38">A11</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="projpl_1.html#R3" title="PROJPL_1:pred.3">on</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self" title="PROJPL_1:def.3">Def3</a></i>;<br/><a NAME="E10:31_1_1_1_1_1"/><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">D</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="projpl_1.html#E3:31_1_1_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="projpl_1.html#T20" target="_self" title="PROJPL_1:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">p</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E12:31_1_1_1_1_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">q</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E14:31_1_1_1_1_1"/><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">r</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E16:31_1_1_1_1_1"/><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><a NAME="E17:31_1_1_1_1_1"/>
( <font color="Maroon" title="c10">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c11">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c12">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E12:31_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="projpl_1.html#E14:31_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="projpl_1.html#E16:31_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self" title="PROJPL_1:def.2">Def2</a></i>;<br/><a NAME="E18:31_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E43">A16</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/><a NAME="E19:31_1_1_1_1_1"/>
<font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">r</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="projpl_1.html#E1:31_1_1_1_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="projpl_1.html#E9:31_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="projpl_1.html#E10:31_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="projpl_1.html#E12:31_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="projpl_1.html#E14:31_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="projpl_1.html#E16:31_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="projpl_1.html#T21" target="_self" title="PROJPL_1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E20:31_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E18:31_1_1_1_1_1"><i><font color="Green" title="E43">A16</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" title="E34">A17</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c2">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:31_1_1_1_1_2"/><i><font color="Green" title="E35">A18</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">C</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:31_1_1_1_1_2"/><i><font color="Green" title="E36">A19</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">D</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:31_1_1_1_1_2"/><i><font color="Green" title="E37">A20</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><a NAME="E8:31_1_1_1_1_2"/>
( <font color="Maroon" title="c4">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c4">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c4">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">D</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1_1_1_1_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_2"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/><a NAME="E9:31_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E38">A21</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="projpl_1.html#R3" title="PROJPL_1:pred.3">on</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self" title="PROJPL_1:def.3">Def3</a></i>;<br/><a NAME="E10:31_1_1_1_1_2"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="projpl_1.html#R5" title="PROJPL_1:pred.5">is_a_quadrangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="projpl_1.html#T13" target="_self" title="PROJPL_1:th.13">Th13</a></i>;<br/><a NAME="E11:31_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E39">A22</font></i>: 
<font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">D</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">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" title="E35">A18</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_2"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="projpl_1.html#T20" target="_self" title="PROJPL_1:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">p</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E13:31_1_1_1_1_2"/><i><font color="Green" title="E40">A23</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">q</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E15:31_1_1_1_1_2"/><i><font color="Green" title="E41">A24</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">r</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E17:31_1_1_1_1_2"/><i><font color="Green" title="E42">A25</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><a NAME="E18:31_1_1_1_1_2"/>
( <font color="Maroon" title="c10">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c11">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c12">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E13:31_1_1_1_1_2"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_2"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_2"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self" title="PROJPL_1:def.2">Def2</a></i>;<br/><a NAME="E19:31_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E43">A26</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/><a NAME="E20:31_1_1_1_1_2"/>
<font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">r</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="projpl_1.html#E1:31_1_1_1_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="projpl_1.html#E9:31_1_1_1_1_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="projpl_1.html#E11:31_1_1_1_1_2"><i><font color="Green" title="E39">A22</font></i></a>, <a class="txt" href="projpl_1.html#E13:31_1_1_1_1_2"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_2"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_2"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="projpl_1.html#T21" target="_self" title="PROJPL_1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E21:31_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E19:31_1_1_1_1_2"><i><font color="Green" title="E43">A26</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" title="E34">A27</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="projpl_1.html#NR1" title="PROJPL_1:NR.1">|'</a> <font color="Maroon" title="c2">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:31_1_1_1_1_3"/><i><font color="Green" title="E35">A28</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">C</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:31_1_1_1_1_3"/><i><font color="Green" title="E36">A29</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">D</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:31_1_1_1_1_3"/><i><font color="Green" title="E37">A30</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><a NAME="E8:31_1_1_1_1_3"/>
( <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">D</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31_1_1_1_1_3"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_3"><i><font color="Green" title="E36">A29</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_3"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/><a NAME="E9:31_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E38">A31</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="projpl_1.html#R3" title="PROJPL_1:pred.3">on</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self" title="PROJPL_1:def.3">Def3</a></i>;<br/><a NAME="E10:31_1_1_1_1_3"/>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> <a href="projpl_1.html#R5" title="PROJPL_1:pred.5">is_a_quadrangle</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E2:31_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="projpl_1.html#T13" target="_self" title="PROJPL_1:th.13">Th13</a></i>;<br/><a NAME="E11:31_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E39">A32</font></i>: 
<font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">D</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">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" title="E35">A28</font></i></a>, <a class="txt" href="projpl_1.html#E5:31_1_1_1_1_3"><i><font color="Green" title="E36">A29</font></i></a>, <a class="txt" href="projpl_1.html#E7:31_1_1_1_1_3"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="projpl_1.html#T20" target="_self" title="PROJPL_1:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">p</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E13:31_1_1_1_1_3"/><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">q</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E15:31_1_1_1_1_3"/><i><font color="Green" title="E41">A34</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">r</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E17:31_1_1_1_1_3"/><i><font color="Green" title="E42">A35</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="projpl_1.html#R2" title="PROJPL_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c9">D</font>
 <b>by </b><i><a class="txt" href="projpl_1.html#E3:31"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><a NAME="E18:31_1_1_1_1_3"/>
( <font color="Maroon" title="c10">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c11">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c12">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E13:31_1_1_1_1_3"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_3"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_3"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="projpl_1.html#D2" target="_self" title="PROJPL_1:def.2">Def2</a></i>;<br/><a NAME="E19:31_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E43">A36</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/><a NAME="E20:31_1_1_1_1_3"/>
<font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">r</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="projpl_1.html#E1:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="projpl_1.html#E1:31_1_1_1_1_3"><i><font color="Green" title="E34">A27</font></i></a>, <a class="txt" href="projpl_1.html#E9:31_1_1_1_1_3"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="projpl_1.html#E11:31_1_1_1_1_3"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="projpl_1.html#E13:31_1_1_1_1_3"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="projpl_1.html#E15:31_1_1_1_1_3"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="projpl_1.html#E17:31_1_1_1_1_3"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="projpl_1.html#T21" target="_self" title="PROJPL_1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E21:31_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">P</font> )
 <b>by </b><i><a class="txt" href="projpl_1.html#E19:31_1_1_1_1_3"><i><font color="Green" title="E43">A36</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" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">P</font> )
 ;<br/>


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


</div>
