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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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; 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="b11">S</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="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="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/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E16">A1</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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; 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="b11">S</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="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="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/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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; 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="b4">S</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="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="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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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; 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="c12">S</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="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="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/>



<b>assume </b><b>that </b><br/><a NAME="E2:21"/><i><font color="Green" title="E17">A2</font></i>: 
( 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; 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="c12">S</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="c12">S</font> )
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green" title="E18">A3</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E4:21"/><i><font color="Green" title="E19">A4</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c17">q'</font> = <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c18">r'</font> = <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c19">s'</font> = <font color="Maroon" title="c5">s</font>, <font color="Maroon" title="c20">a'</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c21">b'</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c22">c'</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="E6:21"/>
( <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> )
 
<b>by </b><i><a class="txt" href="incproj.html#E3:21"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E7:21"/><b>then </b><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c20">a'</font>,<font color="Maroon" title="c21">b'</font>,<font color="Maroon" title="c22">c'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="incproj.html#E4:21"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/>
<a NAME="E8:21"/><i><font color="Green" title="E21">A6</font></i>: 
( <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> &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="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> &amp; <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> &amp; <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> &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="c11">R</font> &amp; <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> &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="c11">R</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="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> &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="c12">S</font> &amp; <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> &amp; <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> &amp; <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> &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="c14">B</font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E3:21"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E9:21"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c18">r'</font>,<font color="Maroon" title="c21">b'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c19">s'</font>,<font color="Maroon" title="c17">q'</font>,<font color="Maroon" title="c21">b'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c19">s'</font>,<font color="Maroon" title="c20">a'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c18">r'</font>,<font color="Maroon" title="c17">q'</font>,<font color="Maroon" title="c20">a'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font>,<font color="Maroon" title="c22">c'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c18">r'</font>,<font color="Maroon" title="c19">s'</font>,<font color="Maroon" title="c22">c'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/>
<div><i><font color="Green" title="E23">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c18">r'</font>,<font color="Maroon" title="c17">q'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:21_1"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
 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> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E3:21_1"/>
<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>
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>thus </b><a NAME="E4:21_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="incproj.html#E2:21_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E24">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c18">r'</font>,<font color="Maroon" title="c19">s'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:21_2"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
 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> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E3:21_2"/>
<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>
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>thus </b><a NAME="E4:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="incproj.html#E2:21_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E25">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3"/>
<font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c19">s'</font>,<font color="Maroon" title="c17">q'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:21_3"/><b>then </b><i><font color="Green" title="E26">A13</font></i>: 
 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> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E3:21_3"/>
<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>
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>thus </b><a NAME="E4:21_3"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="incproj.html#E2:21_3"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_4"/>
<font color="Maroon" title="c18">r'</font>,<font color="Maroon" title="c19">s'</font>,<font color="Maroon" title="c17">q'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:21_4"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
 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> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E3:21_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>
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>thus </b><a NAME="E4:21_4"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E2:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="incproj.html#E8:21"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="incproj.html#E2:21_4"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:21"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="incproj.html#E7:21"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="incproj.html#E9:21"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="incproj.html#E10:21"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="incproj.html#E11:21"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="incproj.html#E12:21"><i><font color="Green" title="E25">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
