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

<span class="kw">let </span><font color="Maroon" title="c1">IPP</font> be   <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</a>   <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a>  <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">d</font>, <font color="Olive" title="b7">pp9</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><br/> for <font color="Olive" title="b8">A</font>, <font color="Olive" title="b9">C</font>, <font color="Olive" title="b10">B</font>, <font color="Olive" title="b11">O</font>, <font color="Olive" title="b12">Q</font>, <font color="Olive" title="b13">O1</font>, <font color="Olive" title="b14">O2</font>, <font color="Olive" title="b15">O3</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>  st not <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">A</font> &amp; not <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">C</font> &amp; not <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">C</font> &amp; not <font color="Olive" title="b3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">A</font> &amp; not <font color="Olive" title="b8">A</font>,<font color="Olive" title="b10">B</font>,<font color="Olive" title="b9">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Olive" title="b10">B</font>,<font color="Olive" title="b9">C</font>,<font color="Olive" title="b11">O</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">p</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="b8">A</font> &amp; <font color="Olive" title="b6">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b10">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="b4">c</font>,<font color="Olive" title="b6">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b9">C</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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="b11">O</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b4">c</font>,<font color="Olive" title="b7">pp9</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="b12">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="b1">a</font>,<font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">p</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">O1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b3">q</font>,<font color="Olive" title="b5">p</font>,<font color="Olive" title="b7">pp9</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">O2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">d</font>,<font color="Olive" title="b7">pp9</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="b15">O3</font> holds <br/>( <font color="Olive" title="b12">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">A</font> &amp; <font color="Olive" title="b12">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">C</font> &amp; not <font color="Olive" title="b3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b12">Q</font> &amp; not <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b12">Q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">d</font>, <font color="Maroon" title="c8">pp9</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">B</font>, <font color="Olive" title="b4">O</font>, <font color="Olive" title="b5">Q</font>, <font color="Olive" title="b6">O1</font>, <font color="Olive" title="b7">O2</font>, <font color="Olive" title="b8">O3</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>  st not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b2">C</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font> &amp; not <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">B</font>,<font color="Olive" title="b2">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Olive" title="b3">B</font>,<font color="Olive" title="b2">C</font>,<font color="Olive" title="b4">O</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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</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="b1">A</font> &amp; <font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b2">C</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">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="b4">O</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">pp9</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="b5">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="c2">a</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">p</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="b6">O1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">pp9</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="b7">O2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">pp9</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">O3</font> holds <br/>( <font color="Olive" title="b5">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">Q</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="Olive" title="b5">Q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">A</font>, <font color="Maroon" title="c10">C</font>, <font color="Maroon" title="c11">B</font>, <font color="Maroon" title="c12">O</font>, <font color="Maroon" title="c13">Q</font>, <font color="Maroon" title="c14">O1</font>, <font color="Maroon" title="c15">O2</font>, <font color="Maroon" title="c16">O3</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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="c9">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="c10">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="c10">C</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> &amp; not <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c10">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; not <font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c12">O</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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">A</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="c11">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c10">C</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">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="c12">O</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">pp9</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="c13">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="c2">a</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">p</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="c14">O1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">pp9</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="c15">O2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">pp9</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="c16">O3</font> implies ( <font color="Maroon" title="c13">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c13">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">Q</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">Q</font> ) )</span><br/>































<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
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="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
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="c10">C</font>
 <span class="kw">and </span><br/><a NAME="E3:21"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
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="c10">C</font>
 <span class="kw">and </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E5:21"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
not <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c10">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 <span class="kw">and </span><br/><a NAME="E6:21"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
not <font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c12">O</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a> 
 <span class="kw">and </span><br/><a NAME="E7:21"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:21"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">and </span><br/><a NAME="E9:21"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E10:21"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E11:21"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">B</font>
 <span class="kw">and </span><br/><a NAME="E12:21"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">and </span><br/><a NAME="E13:21"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">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="c12">O</font>
 <span class="kw">and </span><br/><a NAME="E14:21"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">pp9</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="c13">Q</font>
 <span class="kw">and </span><br/><a NAME="E15:21"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">p</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="c14">O1</font>
 <span class="kw">and </span><br/><a NAME="E16:21"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">pp9</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="c15">O2</font>
 <span class="kw">and </span><br/><a NAME="E17:21"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">pp9</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="c16">O3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c13">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">Q</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">Q</font> )</span><br/>

<a NAME="E18:21"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E19:21"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E20:21"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E21:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E22:21"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c8">pp9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E23:21"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E24:21"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c8">pp9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E25:21"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E26:21"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
<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">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E27:21"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
<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">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E28:21"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E29:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c12">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E30:21"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E31:21"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E32:21"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
<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">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E33:21"/><span class="lab"><font color="Green" title="E47">A33</font></span>: 
<font color="Maroon" title="c8">pp9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E34:21"/><span class="lab"><font color="Green" title="E48">A34</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E35:21"/><span class="lab"><font color="Green" title="E49">A35</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E36:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A36</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E37:21"/><span class="lab"><font color="Green" title="E51">A37</font></span>: 
<font color="Maroon" title="c8">pp9</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="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1"/>
not <font color="Maroon" title="c8">pp9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A38</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E3:21_1"/>
( <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">O</font> &amp; <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E38:21"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E39:21"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<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">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E40:21"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">pp9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_2"/>
not <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">pp9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E38:21"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E41:21"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E42:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A43</font></span>: 
<font color="Maroon" title="c16">O3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E38:21"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E43:21"/><span class="lab"><font color="Green" title="E57">A44</font></span>: 
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">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3"/>
<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">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A45</font></span>: 
<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">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E42:21"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E3:21_3"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E4:21_3"/>
( <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O2</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="c12">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E5:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E2:21_3"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E3:21_3"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E44:21"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E45:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A47</font></span>: 
<font color="Maroon" title="c15">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E46:21"/><span class="lab"><font color="Green" title="E59">A48</font></span>: 
<font color="Maroon" title="c12">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E38:21"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E47:21"/>
not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_4"/>
<font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pp9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E45:21"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E46:21"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E48:21"/>
( <font color="Maroon" title="c13">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c13">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C</font> &amp; not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">Q</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">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E38:21"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E37:21"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
