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<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="firebrick">::  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">o</font>, <font color="Olive" title="b6">o99</font>, <font color="Olive" title="b7">d</font>, <font color="Olive" title="b8">o9</font>, <font color="Olive" title="b9">oo9</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="b10">A</font>, <font color="Olive" title="b11">C</font>, <font color="Olive" title="b12">B</font>, <font color="Olive" title="b13">O</font>, <font color="Olive" title="b14">Q</font>, <font color="Olive" title="b15">O1</font>, <font color="Olive" title="b16">O2</font>, <font color="Olive" title="b17">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="b10">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="b11">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="b12">B</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="b11">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="b10">A</font> &amp;  not <font color="Olive" title="b10">A</font>,<font color="Olive" title="b12">B</font>,<font color="Olive" title="b11">C</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; <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">o</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">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="b5">o</font>,<font color="Olive" title="b6">o99</font>,<font color="Olive" title="b7">d</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">B</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b4">c</font>,<font color="Olive" title="b7">d</font>,<font color="Olive" title="b8">o9</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">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="b7">d</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">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="b9">oo9</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="b14">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="b5">o</font>,<font color="Olive" title="b8">o9</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">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="b2">b</font>,<font color="Olive" title="b8">o9</font>,<font color="Olive" title="b9">oo9</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="b16">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="b5">o</font>,<font color="Olive" title="b9">oo9</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="b17">O3</font> &amp; <font color="Olive" title="b3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b13">O</font> holds <br/>(  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="b14">Q</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="b14">Q</font> &amp; <font color="Olive" title="b10">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b14">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">o</font>, <font color="Maroon" title="c7">o99</font>, <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">o9</font>, <font color="Maroon" title="c10">oo9</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="firebrick">::  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="b3">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="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; <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; <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">o</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; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">o99</font>,<font color="Maroon" title="c8">d</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="b3">B</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">o9</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="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="c8">d</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="c10">oo9</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="c6">o</font>,<font color="Maroon" title="c9">o9</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="c3">b</font>,<font color="Maroon" title="c9">o9</font>,<font color="Maroon" title="c10">oo9</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="c6">o</font>,<font color="Maroon" title="c10">oo9</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="b8">O3</font> &amp; <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">O</font> holds <br/>(  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> &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; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">Q</font> )</span><br/><span class="kw">let </span><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">O</font>, <font color="Maroon" title="c15">Q</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="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="c4">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="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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</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">o</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="c11">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">o</font>,<font color="Maroon" title="c7">o99</font>,<font color="Maroon" title="c8">d</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="c13">B</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">o9</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">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="c8">d</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">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="c10">oo9</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">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="c6">o</font>,<font color="Maroon" title="c9">o9</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">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="c3">b</font>,<font color="Maroon" title="c9">o9</font>,<font color="Maroon" title="c10">oo9</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="c17">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="c6">o</font>,<font color="Maroon" title="c10">oo9</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="c18">O3</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="c14">O</font> implies (  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="c15">Q</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="c15">Q</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="c15">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="E16">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="c11">A</font>
 <span class="kw">and </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E17">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="c12">C</font>
 <span class="kw">and </span><br/><a NAME="E3:21"/><span class="lab"><font color="Green" title="E18">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="c13">B</font>
 <span class="kw">and </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E19">A4</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="c12">C</font>
 <span class="kw">and </span><br/><a NAME="E5:21"/><span class="lab"><font color="Green" title="E20">A5</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="c11">A</font>
 <span class="kw">and </span><br/><a NAME="E6:21"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 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> 
 <span class="kw">and </span><br/><a NAME="E7:21"/><span class="lab"><font color="Green" title="E22">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="E23">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="E24">A9</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">o</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="c11">A</font>
 <span class="kw">and </span><br/><a NAME="E10:21"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">o99</font>,<font color="Maroon" title="c8">d</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="c13">B</font>
 <span class="kw">and </span><br/><a NAME="E11:21"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">o9</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">C</font>
 <span class="kw">and </span><br/><a NAME="E12:21"/><span class="lab"><font color="Green" title="E27">A12</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="c8">d</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">O</font>
 <span class="kw">and </span><br/><a NAME="E13:21"/><span class="lab"><font color="Green" title="E28">A13</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="c10">oo9</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">Q</font>
 <span class="kw">and </span><br/><a NAME="E14:21"/><span class="lab"><font color="Green" title="E29">A14</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="c6">o</font>,<font color="Maroon" title="c9">o9</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">O1</font>
 <span class="kw">and </span><br/><a NAME="E15:21"/><span class="lab"><font color="Green" title="E30">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="c3">b</font>,<font color="Maroon" title="c9">o9</font>,<font color="Maroon" title="c10">oo9</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="c17">O2</font>
 <span class="kw">and </span><br/><a NAME="E16:21"/><span class="lab"><font color="Green" title="E31">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="c6">o</font>,<font color="Maroon" title="c10">oo9</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="c18">O3</font>
 <span class="kw">and </span><br/><a NAME="E17:21"/><span class="lab"><font color="Green" title="E32">A17</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">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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="c15">Q</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="c15">Q</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="c15">Q</font> )</span><br/>

<a NAME="E18:21"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E19:21"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
( <font color="Maroon" title="c8">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="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E20:21"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E21:21"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E22:21"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E23:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E24:21"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E25:21"/><span class="lab"><font color="Green" title="E40">A25</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="c10">oo9</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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">oo9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E41">A26</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="c18">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E3:21_1"/>
( <font color="Maroon" title="c4">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="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E4:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<a NAME="E5:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:21"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
<font color="Maroon" title="c10">oo9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E27:21"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E28:21"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c9">o9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E29:21"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c10">oo9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E30:21"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c10">oo9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E31:21"/><span class="lab"><font color="Green" title="E46">A32</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="c15">Q</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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E32:21"/><span class="lab"><font color="Green" title="E47">A33</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="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E33:21"/><span class="lab"><font color="Green" title="E48">A34</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="c16">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E34:21"/><span class="lab"><font color="Green" title="E49">A35</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="c14">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E35:21"/><span class="lab"><font color="Green" title="E50">A36</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E36:21"/>
<font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E37:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A37</font></span>: 
<font color="Maroon" title="c6">o</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="#E6:21"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E38:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A38</font></span>: 
<font color="Maroon" title="c9">o9</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="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<a NAME="E39:21"/><span class="lab"><font color="Green" title="E53">A39</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="c15">Q</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"/>
<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">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A40</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="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<a NAME="E3:21_2"/><span class="lab"><font color="Green" title="E55">A41</font></span>: 
<font color="Maroon" title="c9">o9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E4:21_2"/>
( <font color="Maroon" title="c9">o9</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="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:21"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E3:21_2"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E40:21"/><span class="lab"><font color="Green" title="E54">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="c15">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a></span>;<br/>
<a NAME="E41:21"/><span class="lab"><font color="Green" title="E55">A43</font></span>: 
<font color="Maroon" title="c9">o9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E42:21"/><span class="lab"><font color="Green" title="E56">A44</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">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E43:21"/><span class="lab"><font color="Green" title="E57">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="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E44:21"/><span class="lab"><font color="Green" title="E58">A46</font></span>: 
<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="21_3"><span class="kw">proof </span></a><div class="add">

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

<a NAME="E2:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E42:21"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<a NAME="E3:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E45:21"/><span class="lab"><font color="Green" title="E59">A47</font></span>: 
( <font color="Maroon" title="c6">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="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E46:21"/><span class="lab"><font color="Green" title="E60">A48</font></span>: 
( <font color="Maroon" title="c10">oo9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">Q</font> &amp; <font color="Maroon" title="c10">oo9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incsp_1.html#T1" title="INCSP_1:th.1">INCSP_1:1</a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E47:21"/><span class="lab"><font color="Green" title="E61">A49</font></span>: 
<font color="Maroon" title="c9">o9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E48:21"/><span class="lab"><font color="Green" title="E62">A50</font></span>: 
( <font color="Maroon" title="c10">oo9</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="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E49:21"/><span class="lab"><font color="Green" title="E63">A51</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E50:21"/><span class="lab"><font color="Green" title="E64">A52</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="incsp_1.html#T2" title="INCSP_1:th.2">INCSP_1:2</a></span>;<br/>
<a NAME="E51:21"/>
<font color="Maroon" title="c9">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</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"/>
 not <font color="Maroon" title="c9">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c16">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E42:21"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E49:21"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</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="#E3:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E49:21"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E52:21"/><span class="kw">then </span>
<font color="Maroon" title="c14">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="txt" href="#E42:21"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E49:21"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<a NAME="E53:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A53</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="c10">oo9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<a NAME="E54:21"/><span class="lab"><font color="Green" title="E66">A54</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="c15">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_5"/>
<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">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_5"/><span class="kw">then </span>
<font color="Maroon" title="c15">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E48:21"><span class="lab"><font color="Green" title="E62">A50</font></span></a>, <a class="txt" href="#E53:21"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="txt" href="#E45:21"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="txt" href="#E37:21"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E55:21"/>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">o9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E56:21"/><span class="kw">then </span>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">oo9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E44:21"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E57:21"/>
(  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="c15">Q</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="c15">Q</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="c15">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E46:21"><span class="lab"><font color="Green" title="E60">A48</font></span></a>, <a class="txt" href="#E45:21"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="txt" href="#E54:21"><span class="lab"><font color="Green" title="E66">A54</font></span></a>, <a class="ref" href="incproj.html#D4" title="INCPROJ:def.4">INCPROJ:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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