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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="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">c</font>, <font color="Olive" title="b4">o</font>, <font color="Olive" title="b5">o99</font>, <font color="Olive" title="b6">d</font>, <font color="Olive" title="b7">o9</font>, <font color="Olive" title="b8">oo9</font>, <font color="Olive" title="b9">q</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">Q</font>, <font color="Olive" title="b14">O</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="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b13">Q</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="b10">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b13">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="b3">c</font>,<font color="Olive" title="b4">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="b4">o</font>,<font color="Olive" title="b5">o99</font>,<font color="Olive" title="b6">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="b3">c</font>,<font color="Olive" title="b6">d</font>,<font color="Olive" title="b7">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="b6">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="b14">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="b3">c</font>,<font color="Olive" title="b8">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="b13">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="b4">o</font>,<font color="Olive" title="b7">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="b7">o9</font>,<font color="Olive" title="b8">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="b4">o</font>,<font color="Olive" title="b8">oo9</font>,<font color="Olive" title="b9">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="b9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b14">O</font> holds <br/>( not <font color="Olive" title="b9">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="b9">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b13">Q</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="b9">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">c</font>, <font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">o99</font>, <font color="Maroon" title="c7">d</font>, <font color="Maroon" title="c8">o9</font>, <font color="Maroon" title="c9">oo9</font>, <font color="Maroon" title="c10">q</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>; <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">Q</font>, <font color="Olive" title="b5">O</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="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</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="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">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="c4">c</font>,<font color="Maroon" title="c5">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="c5">o</font>,<font color="Maroon" title="c6">o99</font>,<font color="Maroon" title="c7">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="c4">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">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="c7">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="b5">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="c4">c</font>,<font color="Maroon" title="c9">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="b4">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="c5">o</font>,<font color="Maroon" title="c8">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="c8">o9</font>,<font color="Maroon" title="c9">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="c5">o</font>,<font color="Maroon" title="c9">oo9</font>,<font color="Maroon" title="c10">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="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">O</font> holds <br/>( not <font color="Maroon" title="c10">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="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> )</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">Q</font>, <font color="Maroon" title="c15">O</font>, <font color="Maroon" title="c16">O1</font>, <font color="Maroon" title="c17">O2</font>, <font color="Maroon" title="c18">O3</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>; <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="c11">A</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; not <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c12">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">Q</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">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="c5">o</font>,<font color="Maroon" title="c6">o99</font>,<font color="Maroon" title="c7">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="c4">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">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="c7">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="c15">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="c4">c</font>,<font color="Maroon" title="c9">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="c14">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="c5">o</font>,<font color="Maroon" title="c8">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="c8">o9</font>,<font color="Maroon" title="c9">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="c5">o</font>,<font color="Maroon" title="c9">oo9</font>,<font color="Maroon" title="c10">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="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> implies ( not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> ) )</span><br/>



































<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E14">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="E2:20"/><span class="lab"><font color="Green" title="E15">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="E3:20"/><span class="lab"><font color="Green" title="E16">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="E4:20"/><span class="lab"><font color="Green" title="E17">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="E5:20"/><span class="lab"><font color="Green" title="E18">A5</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="c14">Q</font>
 <span class="kw">and </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E19">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="E7:20"/><span class="lab"><font color="Green" title="E20">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="E8:20"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">Q</font>
 <span class="kw">and </span><br/><a NAME="E9:20"/><span class="lab"><font color="Green" title="E22">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="c4">c</font>,<font color="Maroon" title="c5">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="E10:20"/><span class="lab"><font color="Green" title="E23">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="c5">o</font>,<font color="Maroon" title="c6">o99</font>,<font color="Maroon" title="c7">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="E11:20"/><span class="lab"><font color="Green" title="E24">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="c4">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">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="E12:20"/><span class="lab"><font color="Green" title="E25">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="c7">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="c15">O</font>
 <span class="kw">and </span><br/><a NAME="E13:20"/><span class="lab"><font color="Green" title="E26">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="c4">c</font>,<font color="Maroon" title="c9">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="c14">Q</font>
 <span class="kw">and </span><br/><a NAME="E14:20"/><span class="lab"><font color="Green" title="E27">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="c5">o</font>,<font color="Maroon" title="c8">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="E15:20"/><span class="lab"><font color="Green" title="E28">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="c8">o9</font>,<font color="Maroon" title="c9">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="E16:20"/><span class="lab"><font color="Green" title="E29">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="c5">o</font>,<font color="Maroon" title="c9">oo9</font>,<font color="Maroon" title="c10">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="E17:20"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font>
 ; <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="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> )</span><br/>

<a NAME="E18:20"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E19:20"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
( <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E20:20"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E21:20"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c9">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="#E15:20"><span class="lab"><font color="Green" title="E28">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="E22:20"/><span class="lab"><font color="Green" title="E35">A22</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> &amp; <font color="Maroon" title="c9">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="#E15:20"><span class="lab"><font color="Green" title="E28">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="E23:20"/><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c9">oo9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E24:20"/><span class="lab"><font color="Green" title="E37">A24</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="#E15:20"><span class="lab"><font color="Green" title="E28">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="E25:20"/><span class="lab"><font color="Green" title="E38">A25</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E26:20"/><span class="lab"><font color="Green" title="E39">A26</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E27:20"/><span class="lab"><font color="Green" title="E40">A27</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="c12">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E28:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A28</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E29:20"/><span class="lab"><font color="Green" title="E42">A29</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="c13">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E30:20"/><span class="lab"><font color="Green" title="E43">A30</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">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E31:20"/><span class="lab"><font color="Green" title="E44">A31</font></span>: 
<font color="Maroon" title="c9">oo9</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E32:20"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
<font color="Maroon" title="c8">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="#E15:20"><span class="lab"><font color="Green" title="E28">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="E33:20"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E29">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="E34:20"/><span class="lab"><font color="Green" title="E47">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="#E14:20"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E35:20"/><span class="lab"><font color="Green" title="E48">A35</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="c15">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E36:20"/><span class="lab"><font color="Green" title="E49">A36</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="c15">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E37:20"/><span class="lab"><font color="Green" title="E50">A37</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="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1"/>
not <font color="Maroon" title="c16">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 ; <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="E2:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E37">A24</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="E3:20_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E41">A28</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="E38:20"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
( <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c9">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="#E16:20"><span class="lab"><font color="Green" title="E29">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="E39:20"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c8">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="#E11:20"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E40:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">o9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E41:20"/><span class="lab"><font color="Green" title="E54">A41</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">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_2"/>
not <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font>
 ; <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="E2:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E38:20"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E33:20"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E3:20_2"/><span class="lab"><font color="Green" title="E56">A43</font></span>: 
<font color="Maroon" title="c8">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="#E15:20"><span class="lab"><font color="Green" title="E28">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="E4:20_2"/>
( <font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</font> &amp; <font color="Maroon" title="c8">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="#E14:20"><span class="lab"><font color="Green" title="E27">A14</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="E5:20_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:20"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E40:20"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E3:20_2"><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>; <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="E42:20"/><span class="lab"><font color="Green" title="E55">A44</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E43:20"/><span class="lab"><font color="Green" title="E56">A45</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E44:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A46</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></span>;<br/>
<a NAME="E45:20"/><span class="lab"><font color="Green" title="E58">A47</font></span>: 
<font color="Maroon" title="c8">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="#E14:20"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E46:20"/><span class="kw">then </span>
<font color="Maroon" title="c8">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E44:20"><span class="lab"><font color="Green" title="E57">A46</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E47:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A48</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">oo9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E43:20"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E39:20"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E48:20"/>
<font color="Maroon" title="c8">o9</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="20_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_3"/>
not <font color="Maroon" title="c8">o9</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="E2:20_3"/><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="c15">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E45:20"><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="E3:20_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E41">A28</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="E49:20"/><span class="kw">then </span>
<font color="Maroon" title="c15">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E39:20"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E50:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A49</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">oo9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E31:20"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E51:20"/><span class="lab"><font color="Green" title="E61">A50</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E52:20"/><span class="lab"><font color="Green" title="E62">A51</font></span>: 
not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_4"/>
<font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 ; <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="E2:20_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E38:20"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E33:20"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E50:20"><span class="lab"><font color="Green" title="E60">A49</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="E3:20_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E51:20"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="txt" href="#E44:20"><span class="lab"><font color="Green" title="E57">A46</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="E53:20"/><span class="lab"><font color="Green" title="E63">A52</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<a NAME="E54:20"/>
not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_5"/>
<font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 ; <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="E2:20_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">oo9</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="#E20:20"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E38:20"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E33:20"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E53:20"><span class="lab"><font color="Green" title="E63">A52</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="E3:20_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E47:20"><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="E55:20"/>
( not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c10">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:20"><span class="lab"><font color="Green" title="E62">A51</font></span></a>, <a class="txt" href="#E41:20"><span class="lab"><font color="Green" title="E54">A41</font></span></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>
