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<span class="kw">let </span><font color="Maroon" title="c1">CPS</font> be   <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</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">p1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">p</font>, <font color="Olive" title="b7">r</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font>  st <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">r2</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">p</font>,<font color="Olive" title="b3">q</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) implies for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span><br/> for <font color="Olive" title="b8">L</font>, <font color="Olive" title="b9">Q</font>, <font color="Olive" title="b10">R</font>, <font color="Olive" title="b11">S</font>, <font color="Olive" title="b12">A</font>, <font color="Olive" title="b13">B</font>, <font color="Olive" title="b14">C</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>  st not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">L</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">L</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> &amp; not <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b10">R</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b10">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b8">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b9">Q</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b4">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b10">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b11">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b12">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b13">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b14">C</font> holds <br/> not <font color="Olive" title="b7">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b14">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">p</font>, <font color="Olive" title="b7">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CPS</font>  st <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b3">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">r2</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b6">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b5">q1</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">p</font>,<font color="Olive" title="b3">q</font>,<font color="Olive" title="b7">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b2">r2</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span><br/> for <font color="Olive" title="b8">L</font>, <font color="Olive" title="b9">Q</font>, <font color="Olive" title="b10">R</font>, <font color="Olive" title="b11">S</font>, <font color="Olive" title="b12">A</font>, <font color="Olive" title="b13">B</font>, <font color="Olive" title="b14">C</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>  st not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">L</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b8">L</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> &amp; not <font color="Olive" title="b4">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b9">Q</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b10">R</font> &amp; not <font color="Olive" title="b3">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b10">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b8">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b9">Q</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b4">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b10">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b6">b</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b11">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b12">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b7">c</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b13">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b14">C</font> holds <br/> not <font color="Olive" title="b7">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b14">C</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">s</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>; <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">L</font>, <font color="Olive" title="b2">Q</font>, <font color="Olive" title="b3">R</font>, <font color="Olive" title="b4">S</font>, <font color="Olive" title="b5">A</font>, <font color="Olive" title="b6">B</font>, <font color="Olive" title="b7">C</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>  st not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> &amp; not <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">Q</font> &amp; not <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">Q</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">R</font> &amp; not <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b1">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b2">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="c7">b</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b3">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b4">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">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="b5">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="c8">c</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b6">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b7">C</font> holds <br/> not <font color="Maroon" title="c8">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b7">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">L</font>, <font color="Maroon" title="c10">Q</font>, <font color="Maroon" title="c11">R</font>, <font color="Maroon" title="c12">S</font>, <font color="Maroon" title="c13">A</font>, <font color="Maroon" title="c14">B</font>, <font color="Maroon" title="c15">C</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>; <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="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> &amp; not <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; not <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font> &amp; not <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c10">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="c7">b</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">R</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c12">S</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">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="c13">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="c8">c</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">B</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C</font> implies  not <font color="Maroon" title="c8">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:25"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
not <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font>
 <span class="kw">and </span><br/><a NAME="E3:25"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
not <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font>
 <span class="kw">and </span><br/><a NAME="E4:25"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:25"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
not <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">and </span><br/><a NAME="E6:25"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">and </span><br/><a NAME="E7:25"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
not <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">and </span><br/><a NAME="E8:25"/><span class="lab"><font color="Green" title="E23">A8</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">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">L</font>
 <span class="kw">and </span><br/><a NAME="E9:25"/><span class="lab"><font color="Green" title="E24">A9</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">a</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">and </span><br/><a NAME="E10:25"/><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="c7">b</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">and </span><br/><a NAME="E11:25"/><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="c7">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">and </span><br/><a NAME="E12:25"/><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="c8">c</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">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="c13">A</font>
 <span class="kw">and </span><br/><a NAME="E13:25"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">and </span><br/><a NAME="E14:25"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">C</font>
 ; <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="c8">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c17">q9</font> = <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c18">r9</font> = <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c19">s9</font> = <font color="Maroon" title="c5">s</font>, <font color="Maroon" title="c20">a9</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c21">b9</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c22">c9</font> = <font color="Maroon" title="c8">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font> ;<br/>
<a NAME="E16:25"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> &amp; <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E17:25"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E32">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_1"/>
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c18">r9</font>,<font color="Maroon" title="c19">s9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">S</font> &amp; contradiction )</span><br/><a NAME="E2:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A18</font></span>: 
 ex <font color="Olive" title="b1">K</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_1"/>
<font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E17:25"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">thus </span><a NAME="E4:25_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E17:25"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E2:25_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</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>
<div><span class="lab"><font color="Green" title="E33">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_2"/>
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c19">s9</font>,<font color="Maroon" title="c17">q9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font> &amp; contradiction )</span><br/><a NAME="E2:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A20</font></span>: 
 ex <font color="Olive" title="b1">K</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_2"/>
<font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E17:25"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">thus </span><a NAME="E4:25_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E17:25"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E2:25_2"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</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="E20:25"/>
<font color="Maroon" title="c6">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E21:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c19">s9</font>,<font color="Maroon" title="c20">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<span class="kw">assume </span><a NAME="E22:25"/><span class="lab"><font color="Green" title="E35">A22</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</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="E23:25"/>
( <font color="Maroon" title="c6">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font> &amp; <font color="Maroon" title="c7">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:25"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></span>;<br/>
<a NAME="E24:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c20">a9</font>,<font color="Maroon" title="c21">b9</font>,<font color="Maroon" title="c22">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E25:25"/><span class="lab"><font color="Green" title="E37">A24</font></span>: 
( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E26:25"/><span class="lab"><font color="Green" title="E38">A25</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_3"/>
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c18">r9</font>,<font color="Maroon" title="c17">q9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">S</font> &amp; contradiction )</span><br/><a NAME="E2:25_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A27</font></span>: 
 ex <font color="Olive" title="b1">K</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_3"/>
<font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E25:25"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">thus </span><a NAME="E4:25_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E25:25"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E2:25_3"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</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>
<div><span class="lab"><font color="Green" title="E40">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_4"/>
<font color="Maroon" title="c18">r9</font>,<font color="Maroon" title="c19">s9</font>,<font color="Maroon" title="c17">q9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font> &amp; contradiction )</span><br/><a NAME="E2:25_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A29</font></span>: 
 ex <font color="Olive" title="b1">K</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_4"/>
<font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E25:25"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">thus </span><a NAME="E4:25_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E25:25"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E2:25_4"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</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="E29:25"/>
<font color="Maroon" title="c6">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E30:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A30</font></span>: 
<font color="Maroon" title="c18">r9</font>,<font color="Maroon" title="c17">q9</font>,<font color="Maroon" title="c20">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:25"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E31:25"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E26">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="E32:25"/>
( <font color="Maroon" title="c7">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">S</font> &amp; <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E26">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="E33:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A32</font></span>: 
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c18">r9</font>,<font color="Maroon" title="c21">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:25"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E34:25"/><span class="lab"><font color="Green" title="E44">A33</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E35:25"/>
( <font color="Maroon" title="c8">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Maroon" title="c4">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E36:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A34</font></span>: 
<font color="Maroon" title="c18">r9</font>,<font color="Maroon" title="c19">s9</font>,<font color="Maroon" title="c22">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:25"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E37:25"/><span class="lab"><font color="Green" title="E46">A35</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E27">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="E38:25"/>
( <font color="Maroon" title="c8">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E27">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="E39:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A36</font></span>: 
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>,<font color="Maroon" title="c22">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:25"><span class="lab"><font color="Green" title="E46">A35</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E40:25"/>
<font color="Maroon" title="c7">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E25">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="E41:25"/><span class="kw">then </span>
<font color="Maroon" title="c19">s9</font>,<font color="Maroon" title="c17">q9</font>,<font color="Maroon" title="c21">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E17:25"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E42:25"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E24:25"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E33:25"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="txt" href="#E21:25"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E30:25"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E36:25"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E27:25"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E18:25"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E19:25"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E28:25"><span class="lab"><font color="Green" title="E40">A28</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/>


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