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

<b>let </b><font color="Maroon" title="c1">IPP</font> be   <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/><b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">B</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E10">A1</font></i>: 
 ex <font color="Olive" title="b1">o</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a>  )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">o</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green" title="E12">A3</font></i>: 
( <font color="Maroon" title="c8">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c8">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">B</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; not <font color="Maroon" title="c9">x</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="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; not <font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> )
 <b>by </b><i><a class="ref" href="projred1.html#T3" target="_self" title="PROJRED1:th.3">Th3</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">C</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E5:13_1"/><i><font color="Green" title="E14">A5</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c10">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">w</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E7:13_1"/><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c12">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="ref" href="projred1.html#T8" target="_self" title="PROJRED1:th.8">Th8</a></i>;<br/><div><i><font color="Green" title="E16">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">u</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/><b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon" title="c13">u</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font>
 ;<br/><div><i><font color="Green" title="E18">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o</font>
 ;<br/><b>consider </b><font color="Maroon" title="c14">D</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green" title="E20">A11</font></i>: 
( <font color="Maroon" title="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D</font> &amp; <font color="Maroon" title="c13">u</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/><b>take </b><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c8">o</font>;<br/><b>take </b><font color="Maroon" title="c16">D</font> = <font color="Maroon" title="c14">D</font>;<br/><b>thus </b><a NAME="E4:13_1_1_1"/>
( <font color="Maroon" title="c15">v</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">D</font> &amp; <font color="Maroon" title="c13">u</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">D</font> &amp; <font color="Maroon" title="c15">v</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">D</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E4:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_1_1"><i><font color="Green" title="E20">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green" title="E19">A12</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">o</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">u</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c14">D</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E3:13_1_1_2"/><i><font color="Green" title="E20">A13</font></i>: 
( <font color="Maroon" title="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D</font> &amp; <font color="Maroon" title="c13">u</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/><a NAME="E4:13_1_1_2"/>
not <font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D</font>
 ;<br/>

<a NAME="E2:13_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">u</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">C</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E5:13_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1_2"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">v</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E6:13_1_1_2"/><i><font color="Green" title="E21">A14</font></i>: 
( <font color="Maroon" title="c15">v</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="c15">v</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E4:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="projred1.html#E3:13_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="projred1.html#E1:13_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="incproj.html#D13" title="INCPROJ:def.13">INCPROJ:def 13</a></i>;<br/><b>take </b><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c15">v</font>;<br/><b>take </b><font color="Maroon" title="c17">D</font> = <font color="Maroon" title="c14">D</font>;<br/><b>thus </b><a NAME="E7:13_1_1_2"/>
( <font color="Maroon" title="c16">v</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">D</font> &amp; <font color="Maroon" title="c13">u</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">D</font> &amp; <font color="Maroon" title="c16">v</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">D</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="projred1.html#E6:13_1_1_2"><i><font color="Green" title="E21">A14</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:13_1_1"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> ex <font color="Olive" title="b2">D</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Olive" title="b1">v</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">D</font> &amp; <font color="Maroon" title="c13">u</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">D</font> &amp; <font color="Olive" title="b1">v</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">D</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="projred1.html#E2:13_1_1"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E17">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">u1</font>, <font color="Maroon" title="c14">u2</font>, <font color="Maroon" title="c15">v1</font>, <font color="Maroon" title="c16">v2</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/><b>let </b><font color="Maroon" title="c17">D1</font>, <font color="Maroon" title="c18">D2</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:13_1_2"/><i><font color="Green" title="E18">A16</font></i>: 
( <font color="Maroon" title="c13">u1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c14">u2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font> )
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green" title="E19">A17</font></i>: 
( <font color="Maroon" title="c15">v1</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c13">u1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c15">v1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c16">v2</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">D2</font> &amp; <font color="Maroon" title="c14">u2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">D2</font> &amp; <font color="Maroon" title="c16">v2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">D2</font> )
 ;<br/><b>assume </b><a NAME="E3:13_1_2"/><i><font color="Green" title="E20">A18</font></i>: 
<font color="Maroon" title="c13">u1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">u2</font>
 ;<br/><a NAME="E4:13_1_2"/><i><font color="Green" title="E21">A19</font></i>: 
not <font color="Maroon" title="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><a NAME="E5:13_1_2"/><i><font color="Green" title="E22">A20</font></i>: 
<font color="Maroon" title="c17">D1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="projred1.html#E3:13_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="projred1.html#E5:13_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="projred1.html#E7:13_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="projred1.html#E2:13_1_2"><i><font color="Green" title="E19">A17</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><b>assume </b><a NAME="E6:13_1_2"/>
<font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v2</font>
 ;<br/><a NAME="E7:13_1_2"/><b>then </b>
<font color="Maroon" title="c17">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">D2</font>
 <b>by </b><i><a class="txt" href="projred1.html#E2:13_1_2"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="projred1.html#E4:13_1_2"><i><font color="Green" title="E21">A19</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/><b>hence </b><a NAME="E8:13_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:13_1_2"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="projred1.html#E2:13_1_2"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="projred1.html#E3:13_1_2"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="projred1.html#E5:13_1_2"><i><font color="Green" title="E22">A20</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c13">p</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>, <font color="Maroon" title="c14">D1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E11:13_1"/><i><font color="Green" title="E18">A21</font></i>: 
( <font color="Maroon" title="c13">p</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D1</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D1</font> &amp; <font color="Maroon" title="c13">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">D1</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">q</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>, <font color="Maroon" title="c16">D2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E13:13_1"/><i><font color="Green" title="E19">A22</font></i>: 
( <font color="Maroon" title="c15">q</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">D2</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">D2</font> &amp; <font color="Maroon" title="c15">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">D2</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">r</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>, <font color="Maroon" title="c18">D3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E15:13_1"/><i><font color="Green" title="E20">A23</font></i>: 
( <font color="Maroon" title="c17">r</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">D3</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">D3</font> &amp; <font color="Maroon" title="c17">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">D3</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">s</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>, <font color="Maroon" title="c20">D4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E17:13_1"/><i><font color="Green" title="E21">A24</font></i>: 
( <font color="Maroon" title="c19">s</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="c12">w</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">D4</font> &amp; <font color="Maroon" title="c5">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">D4</font> &amp; <font color="Maroon" title="c19">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">D4</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projred1.html#E8:13_1"><i><font color="Green" title="E16">A7</font></i></a></i>;<br/><a NAME="E18:13_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="incproj.html#D6" title="INCPROJ:def.6">INCPROJ:def 6</a></i>;<br/><a NAME="E19:13_1"/><b>then </b>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">q</font> &amp; <font color="Maroon" title="c15">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">r</font> &amp; <font color="Maroon" title="c17">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">s</font> &amp; <font color="Maroon" title="c19">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c19">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">q</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">r</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="projred1.html#E9:13_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="txt" href="projred1.html#E11:13_1"><i><font color="Green" title="E18">A21</font></i></a>, <a class="txt" href="projred1.html#E13:13_1"><i><font color="Green" title="E19">A22</font></i></a>, <a class="txt" href="projred1.html#E15:13_1"><i><font color="Green" title="E20">A23</font></i></a>, <a class="txt" href="projred1.html#E17:13_1"><i><font color="Green" title="E21">A24</font></i></a></i>;<br/><a NAME="E20:13_1"/><b>then </b>
<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c17">r</font>,<font color="Maroon" title="c19">s</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D6" title="INCPROJ:def.6">INCPROJ:def 6</a></i>;<br/><b>hence </b><a NAME="E21:13_1"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a>  )
 <b>by </b><i><a class="txt" href="projred1.html#E11:13_1"><i><font color="Green" title="E18">A21</font></i></a>, <a class="txt" href="projred1.html#E13:13_1"><i><font color="Green" title="E19">A22</font></i></a>, <a class="txt" href="projred1.html#E15:13_1"><i><font color="Green" title="E20">A23</font></i></a>, <a class="txt" href="projred1.html#E17:13_1"><i><font color="Green" title="E21">A24</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a>  )
 <b>by </b><i><a class="txt" href="projred1.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>


</div>
