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

<b>let </b><font color="Maroon" title="c1">IPP</font> be  <a href="incproj.html#V8" title="INCPROJ:attr.8">Fanoian</a> <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</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="b8">A</font>, <font color="Olive" title="b9">B</font>, <font color="Olive" title="b10">C</font>, <font color="Olive" title="b11">Q</font>, <font color="Olive" title="b12">L</font>, <font color="Olive" title="b13">R</font>, <font color="Olive" title="b14">S</font>, <font color="Olive" title="b15">D</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> ex <font color="Olive" title="b16">d</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/>( not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b12">L</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b12">L</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b11">Q</font> &amp; not <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b11">Q</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b13">R</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b13">R</font> &amp; not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b14">S</font> &amp; not <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b14">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</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="Olive" title="b12">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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="Olive" title="b11">Q</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b4">s</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="Olive" title="b13">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">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="Olive" title="b14">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</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="Olive" title="b8">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</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="Olive" title="b9">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">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="Olive" title="b10">C</font> &amp; not <font color="Olive" title="b7">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b10">C</font> &amp; <font color="Olive" title="b6">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b15">D</font> &amp; <font color="Olive" title="b7">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b15">D</font> &amp; <font color="Olive" title="b10">C</font>,<font color="Olive" title="b15">D</font>,<font color="Olive" title="b13">R</font>,<font color="Olive" title="b14">S</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  &amp; <font color="Olive" title="b16">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">A</font> &amp; <font color="Olive" title="b7">c</font>,<font color="Olive" title="b16">d</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  )</span><br/>

<b>consider </b><font color="Maroon" title="c2">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="c3">A</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="E2:15"/><i><font color="Green" title="E12">A1</font></i>: 
not <font color="Maroon" title="c2">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="incproj.html#D11" title="INCPROJ:def.11">INCPROJ:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">c</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="E4:15"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D12" title="INCPROJ:def.12">INCPROJ:def 12</a></i>;<br/>
<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">IPP</font><b> such that </b><br/><a NAME="E6:15"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c2">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="c6">c</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="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">s</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="E8:15"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c8">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="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> )
 <b>by </b><i><a class="ref" href="projred1.html#T8" target="_self" title="PROJRED1:th.8">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Q</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="E10:15"/><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c2">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Q</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="c10">L</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="E12:15"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font> &amp; <font color="Maroon" title="c8">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</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="c11">S</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="E14:15"/><i><font color="Green" title="E18">A7</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">S</font> &amp; <font color="Maroon" title="c2">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">S</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">R</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="E16:15"/><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">R</font> &amp; <font color="Maroon" title="c8">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">R</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E17:15"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c12">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
not <font color="Maroon" title="c12">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_1"/><b>then </b>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</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="E3:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E18:15"/><b>then </b><i><font color="Green" title="E21">A10</font></i>: 
not <font color="Maroon" title="c2">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">R</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E19:15"/><i><font color="Green" title="E22">A11</font></i>: 
not <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E20:15"/><i><font color="Green" title="E23">A12</font></i>: 
not <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">S</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E21:15"/><i><font color="Green" title="E24">A13</font></i>: 
not <font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon" title="c5">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_2"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</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="E3:15_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E22:15"/><i><font color="Green" title="E25">A14</font></i>: 
not <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3"/>
<font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_3"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</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="E3:15_3"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E23:15"/><i><font color="Green" title="E26">A15</font></i>: 
not <font color="Maroon" title="c6">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E24:15"/>
not <font color="Maroon" title="c6">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_4"/>
<font color="Maroon" title="c6">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_4"/><b>then </b>
<font color="Maroon" title="c4">p</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#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</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="E3:15_4"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c13">a</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="E26:15"/><i><font color="Green" title="E27">A16</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c13">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred1.html#E23:15"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="incproj.html#D13" title="INCPROJ:def.13">INCPROJ:def 13</a></i>;<br/>
<a NAME="E27:15"/><i><font color="Green" title="E28">A17</font></i>: 
not <font color="Maroon" title="c6">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">R</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E28:15"/><i><font color="Green" title="E29">A18</font></i>: 
not <font color="Maroon" title="c6">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">S</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">b</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="E30:15"/><i><font color="Green" title="E30">A19</font></i>: 
( <font color="Maroon" title="c14">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">R</font> &amp; <font color="Maroon" title="c14">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">S</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred1.html#E27:15"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="incproj.html#D13" title="INCPROJ:def.13">INCPROJ:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">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="E32:15"/><i><font color="Green" title="E31">A20</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font> &amp; <font color="Maroon" title="c14">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E33:15"/><i><font color="Green" title="E32">A21</font></i>: 
not <font color="Maroon" title="c8">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E23:15"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E34:15"/><i><font color="Green" title="E33">A22</font></i>: 
<font color="Maroon" title="c11">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E28:15"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E35:15"/><b>then </b><i><font color="Green" title="E34">A23</font></i>: 
not <font color="Maroon" title="c2">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E36:15"/><i><font color="Green" title="E35">A24</font></i>: 
not <font color="Maroon" title="c8">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">S</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E28:15"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E37:15"/><i><font color="Green" title="E36">A25</font></i>: 
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">s</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="c10">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">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="c9">Q</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">s</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="c12">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">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="c11">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</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="c3">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c8">s</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="c7">B</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E38:15"/><i><font color="Green" title="E37">A26</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">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="c15">C</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">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="E39:15"/><b>then </b><i><font color="Green" title="E38">A27</font></i>: 
not <font color="Maroon" title="c6">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E18:15"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="projred1.html#E19:15"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="projred1.html#E21:15"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="projred1.html#E33:15"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="projred1.html#E35:15"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="projred1.html#E36:15"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="projred1.html#E37:15"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="incproj.html#D18" title="INCPROJ:def.18">INCPROJ:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">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="E41:15"/><i><font color="Green" title="E39">A28</font></i>: 
( <font color="Maroon" title="c14">b</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="c6">c</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="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E42:15"/><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">s</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E6:15"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred1.html#E8:15"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E23:15"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E43:15"/><i><font color="Green" title="E41">A30</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E34:15"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E44:15"/><i><font color="Green" title="E42">A31</font></i>: 
<font color="Maroon" title="c11">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E45:15"/><i><font color="Green" title="E43">A32</font></i>: 
not <font color="Maroon" title="c13">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">S</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E43:15"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E46:15"/><i><font color="Green" title="E44">A33</font></i>: 
<font color="Maroon" title="c11">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">D</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E47:15"/><i><font color="Green" title="E45">A34</font></i>: 
<font color="Maroon" title="c12">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">D</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E48:15"/><i><font color="Green" title="E46">A35</font></i>: 
<font color="Maroon" title="c12">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">C</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E12:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred1.html#E42:15"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E49:15"/><i><font color="Green" title="E47">A36</font></i>: 
<font color="Maroon" title="c11">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">C</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred1.html#E43:15"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E50:15"/><i><font color="Green" title="E48">A37</font></i>: 
<font color="Maroon" title="c15">C</font>,<font color="Maroon" title="c16">D</font>,<font color="Maroon" title="c12">R</font>,<font color="Maroon" title="c11">S</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred1.html#E39:15"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="projred1.html#E45:15"><i><font color="Green" title="E43">A32</font></i></a>, <a class="txt" href="projred1.html#E46:15"><i><font color="Green" title="E44">A33</font></i></a>, <a class="txt" href="projred1.html#E47:15"><i><font color="Green" title="E45">A34</font></i></a>, <a class="txt" href="projred1.html#E48:15"><i><font color="Green" title="E46">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></i>;<br/>
<a NAME="E51:15"/><i><font color="Green" title="E49">A38</font></i>: 
not <font color="Maroon" title="c14">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E16:15"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred1.html#E17:15"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E52:15"/><b>then </b>
not <font color="Maroon" title="c2">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred1.html#E43:15"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">d</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="E54:15"/><i><font color="Green" title="E50">A39</font></i>: 
( <font color="Maroon" title="c17">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c17">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:15"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred1.html#E44:15"><i><font color="Green" title="E42">A31</font></i></a>, <a class="ref" href="incproj.html#D13" title="INCPROJ:def.13">INCPROJ:def 13</a></i>;<br/>
<a NAME="E55:15"/><i><font color="Green" title="E51">A40</font></i>: 
<font color="Maroon" title="c17">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E10:15"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred1.html#E21:15"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="projred1.html#E26:15"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred1.html#E51:15"><i><font color="Green" title="E49">A38</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green" title="E50">A39</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E56:15"/>
<font color="Maroon" title="c17">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E14:15"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="projred1.html#E30:15"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="projred1.html#E32:15"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred1.html#E49:15"><i><font color="Green" title="E47">A36</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green" title="E50">A39</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E57:15"/><b>then </b>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="projred1.html#E4:15"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred1.html#E39:15"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green" title="E50">A39</font></i></a>, <a class="txt" href="projred1.html#E55:15"><i><font color="Green" title="E51">A40</font></i></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E58:15"/>
 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>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</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="b8">A</font>, <font color="Olive" title="b9">B</font>, <font color="Olive" title="b10">C</font>, <font color="Olive" title="b11">Q</font>, <font color="Olive" title="b12">L</font>, <font color="Olive" title="b13">R</font>, <font color="Olive" title="b14">S</font>, <font color="Olive" title="b15">D</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> ex <font color="Olive" title="b16">d</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/>( not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b12">L</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b12">L</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b11">Q</font> &amp; not <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b11">Q</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b13">R</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b13">R</font> &amp; not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b14">S</font> &amp; not <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b14">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</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="Olive" title="b12">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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="Olive" title="b11">Q</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b4">s</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="Olive" title="b13">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">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="Olive" title="b14">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</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="Olive" title="b8">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</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="Olive" title="b9">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">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="Olive" title="b10">C</font> &amp; not <font color="Olive" title="b7">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b10">C</font> &amp; <font color="Olive" title="b6">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b15">D</font> &amp; <font color="Olive" title="b7">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b15">D</font> &amp; <font color="Olive" title="b10">C</font>,<font color="Olive" title="b15">D</font>,<font color="Olive" title="b13">R</font>,<font color="Olive" title="b14">S</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  &amp; <font color="Olive" title="b16">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">A</font> &amp; <font color="Olive" title="b7">c</font>,<font color="Olive" title="b16">d</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  )
 <b>by </b><i><a class="txt" href="projred1.html#E18:15"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="projred1.html#E19:15"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="projred1.html#E20:15"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="projred1.html#E21:15"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="projred1.html#E22:15"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="projred1.html#E33:15"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="projred1.html#E35:15"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="projred1.html#E36:15"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="projred1.html#E37:15"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="projred1.html#E38:15"><i><font color="Green" title="E37">A26</font></i></a>, <a class="txt" href="projred1.html#E39:15"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="projred1.html#E41:15"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="projred1.html#E50:15"><i><font color="Green" title="E48">A37</font></i></a>, <a class="txt" href="projred1.html#E54:15"><i><font color="Green" title="E50">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
