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

<b>let </b><font color="Maroon" title="c1">IPP</font> be  <a href="incproj.html#V6" title="INCPROJ:attr.6">2-dimensional</a> <a href="incproj.html#V10" title="INCPROJ:attr.10">Desarguesian</a> <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">o</font>, <font color="Maroon" title="c6">o''</font>, <font color="Maroon" title="c7">d</font>, <font color="Maroon" title="c8">o'</font>, <font color="Maroon" title="c9">oo'</font>, <font color="Maroon" title="c10">q</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="c11">A</font>, <font color="Maroon" title="c12">C</font>, <font color="Maroon" title="c13">B</font>, <font color="Maroon" title="c14">Q</font>, <font color="Maroon" title="c15">O</font>, <font color="Maroon" title="c16">O1</font>, <font color="Maroon" title="c17">O2</font>, <font color="Maroon" title="c18">O3</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><a NAME="E1:19"/><i><font color="Green" title="E14">A1</font></i>: 
( not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; not <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c12">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <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="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">o</font> <a href="incproj.html#R3" title="INCPROJ:pred.3">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">o''</font>,<font color="Maroon" title="c7">d</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">o'</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">d</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c15">O</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">oo'</font> <a href="incproj.html#R3" title="INCPROJ:pred.3">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">o'</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c16">O1</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">o'</font>,<font color="Maroon" title="c9">oo'</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c17">O2</font> &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">oo'</font>,<font color="Maroon" title="c10">q</font> <a href="incproj.html#R4" title="INCPROJ:pred.4">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> )
 ;<br/>

<a NAME="E2:19"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
( not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; not <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c12">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <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="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">Q</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="c11">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c6">o''</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</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="c12">C</font> &amp; <font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</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="c12">C</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="c15">O</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="c15">O</font> &amp; <font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</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="c14">Q</font> &amp; <font color="Maroon" title="c9">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</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="c16">O1</font> &amp; <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</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="c16">O1</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="c17">O2</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="c17">O2</font> &amp; <font color="Maroon" title="c9">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font> &amp; <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c9">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D7" title="INCPROJ:def.7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">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="c12">C</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="c14">Q</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="incproj.html#D7" title="INCPROJ:def.7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="incproj.html#D7" title="INCPROJ:def.7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E5:19"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</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="c17">O2</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c9">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c9">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="incproj.html#D7" title="INCPROJ:def.7">INCPROJ:def 7</a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green" title="E20">A7</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c9">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E21">A8</font></i>: 
( <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E9:19"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c5">o</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="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/>
<a NAME="E10:19"/><b>then </b><i><font color="Green" title="E23">A10</font></i>: 
<font color="Maroon" title="c8">o'</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="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E11:19"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/>
<a NAME="E12:19"/><i><font color="Green" title="E25">A12</font></i>: 
<font color="Maroon" title="c16">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
not <font color="Maroon" title="c16">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</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="E3:19_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="projred2.html#E11:19"><i><font color="Green" title="E24">A11</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>
<a NAME="E13:19"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="projred2.html#E11:19"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E14:19"/><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">oo'</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="projred2.html#E10:19"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E15:19"/><i><font color="Green" title="E28">A15</font></i>: 
not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 ;<br/>

<a NAME="E2:19_2"/><b>then </b>
<font color="Maroon" title="c9">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E4:19"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="projred2.html#E7:19"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="projred2.html#E8:19"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="projred2.html#E13:19"><i><font color="Green" title="E26">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="E3:19_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="projred2.html#E3:19"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="projred2.html#E6:19"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="projred2.html#E14:19"><i><font color="Green" title="E27">A14</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>
<a NAME="E16:19"/>
<font color="Maroon" title="c8">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_3"/>
not <font color="Maroon" title="c8">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 ;<br/>

<a NAME="E2:19_3"/><b>then </b>
<font color="Maroon" title="c16">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">O</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</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="E3:19_3"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="projred2.html#E11:19"><i><font color="Green" title="E24">A11</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>
<a NAME="E17:19"/><b>then </b>
<font color="Maroon" title="c15">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E18:19"/><b>then </b><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">oo'</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E19:19"/><i><font color="Green" title="E30">A17</font></i>: 
not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_4"/>
<font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 ;<br/>

<a NAME="E2:19_4"/><b>then </b>
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E6:19"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="projred2.html#E7:19"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="projred2.html#E8:19"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="projred2.html#E18:19"><i><font color="Green" title="E29">A16</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="E3:19_4"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="projred2.html#E9:19"><i><font color="Green" title="E22">A9</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>
<a NAME="E20:19"/><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">o'</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/>
<a NAME="E21:19"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_5"/>
not <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font>
 ;<br/>

<a NAME="E2:19_5"/><b>then </b>
( <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</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="c16">O1</font> &amp; <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</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="c17">O2</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="projred2.html#E5:19"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="projred2.html#E6:19"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="projred2.html#E7:19"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="projred2.html#E8:19"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="incproj.html#D8" title="INCPROJ:def.8">INCPROJ:def 8</a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:19_5"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E12:19"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="projred2.html#E20:19"><i><font color="Green" title="E31">A18</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>hence </b><a NAME="E22:19"/>
( not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</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="c10">q</font> )
 <b>by </b><i><a class="txt" href="projred2.html#E15:19"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="projred2.html#E19:19"><i><font color="Green" title="E30">A17</font></i></a></i>;<br/>


</div>
