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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">o</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">p3</font>, <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">q2</font>, <font color="Olive" title="b7">q3</font>, <font color="Olive" title="b8">r1</font>, <font color="Olive" title="b9">r2</font>, <font color="Olive" title="b10">r3</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">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q1</font> &amp; <font color="Olive" title="b2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q2</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">q3</font> &amp; <font color="Olive" title="b4">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">q3</font> &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b4">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">q1</font>,<font color="Olive" title="b6">q2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">p2</font>,<font color="Olive" title="b4">p3</font>,<font color="Olive" title="b8">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">q2</font>,<font color="Olive" title="b7">q3</font>,<font color="Olive" title="b8">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">p3</font>,<font color="Olive" title="b9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">q1</font>,<font color="Olive" title="b7">q3</font>,<font color="Olive" title="b9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b6">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b4">p3</font>,<font color="Olive" title="b7">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b8">r1</font>,<font color="Olive" title="b9">r2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) implies for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">b2</font>, <font color="Olive" title="b5">a2</font>, <font color="Olive" title="b6">b3</font>, <font color="Olive" title="b7">a3</font>, <font color="Olive" title="b8">r</font>, <font color="Olive" title="b9">s</font>, <font color="Olive" title="b10">t</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="b11">C1</font>, <font color="Olive" title="b12">C2</font>, <font color="Olive" title="b13">C3</font>, <font color="Olive" title="b14">A1</font>, <font color="Olive" title="b15">A2</font>, <font color="Olive" title="b16">A3</font>, <font color="Olive" title="b17">B1</font>, <font color="Olive" title="b18">B2</font>, <font color="Olive" title="b19">B3</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">a1</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">C1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b4">b2</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">C2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b6">b3</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">C3</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">a3</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b10">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b14">A1</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">a3</font>,<font color="Olive" title="b8">r</font>,<font color="Olive" title="b3">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b15">A2</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">a2</font>,<font color="Olive" title="b9">s</font>,<font color="Olive" title="b3">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b16">A3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b10">t</font>,<font color="Olive" title="b4">b2</font>,<font color="Olive" title="b6">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b17">B1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b8">r</font>,<font color="Olive" title="b6">b3</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="b18">B2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b9">s</font>,<font color="Olive" title="b4">b2</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="b19">B3</font> &amp; <font color="Olive" title="b11">C1</font>,<font color="Olive" title="b12">C2</font>,<font color="Olive" title="b13">C3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">a3</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> &amp; <font color="Olive" title="b3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b7">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> holds <br/> ex <font color="Olive" title="b20">O</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b8">r</font>,<font color="Olive" title="b9">s</font>,<font color="Olive" title="b10">t</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="b20">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">p3</font>, <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">q2</font>, <font color="Olive" title="b7">q3</font>, <font color="Olive" title="b8">r1</font>, <font color="Olive" title="b9">r2</font>, <font color="Olive" title="b10">r3</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">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q1</font> &amp; <font color="Olive" title="b2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q2</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">q3</font> &amp; <font color="Olive" title="b4">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">q3</font> &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b4">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">q1</font>,<font color="Olive" title="b6">q2</font>,<font color="Olive" title="b10">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">p2</font>,<font color="Olive" title="b4">p3</font>,<font color="Olive" title="b8">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">q2</font>,<font color="Olive" title="b7">q3</font>,<font color="Olive" title="b8">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">p3</font>,<font color="Olive" title="b9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">q1</font>,<font color="Olive" title="b7">q3</font>,<font color="Olive" title="b9">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b6">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b4">p3</font>,<font color="Olive" title="b7">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b8">r1</font>,<font color="Olive" title="b9">r2</font>,<font color="Olive" title="b10">r3</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">o</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">b2</font>, <font color="Olive" title="b5">a2</font>, <font color="Olive" title="b6">b3</font>, <font color="Olive" title="b7">a3</font>, <font color="Olive" title="b8">r</font>, <font color="Olive" title="b9">s</font>, <font color="Olive" title="b10">t</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="b11">C1</font>, <font color="Olive" title="b12">C2</font>, <font color="Olive" title="b13">C3</font>, <font color="Olive" title="b14">A1</font>, <font color="Olive" title="b15">A2</font>, <font color="Olive" title="b16">A3</font>, <font color="Olive" title="b17">B1</font>, <font color="Olive" title="b18">B2</font>, <font color="Olive" title="b19">B3</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">a1</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">C1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b4">b2</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">C2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">a3</font>,<font color="Olive" title="b6">b3</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">C3</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">a3</font>,<font color="Olive" title="b5">a2</font>,<font color="Olive" title="b10">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b14">A1</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">a3</font>,<font color="Olive" title="b8">r</font>,<font color="Olive" title="b3">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b15">A2</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">a2</font>,<font color="Olive" title="b9">s</font>,<font color="Olive" title="b3">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b16">A3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b10">t</font>,<font color="Olive" title="b4">b2</font>,<font color="Olive" title="b6">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b17">B1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b8">r</font>,<font color="Olive" title="b6">b3</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="b18">B2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b9">s</font>,<font color="Olive" title="b4">b2</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="b19">B3</font> &amp; <font color="Olive" title="b11">C1</font>,<font color="Olive" title="b12">C2</font>,<font color="Olive" title="b13">C3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">a3</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> &amp; <font color="Olive" title="b3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b7">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b3</font> holds <br/> ex <font color="Olive" title="b20">O</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b8">r</font>,<font color="Olive" title="b9">s</font>,<font color="Olive" title="b10">t</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="b20">O</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">b2</font>, <font color="Maroon" title="c6">a2</font>, <font color="Maroon" title="c7">b3</font>, <font color="Maroon" title="c8">a3</font>, <font color="Maroon" title="c9">r</font>, <font color="Maroon" title="c10">s</font>, <font color="Maroon" title="c11">t</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">C1</font>, <font color="Olive" title="b2">C2</font>, <font color="Olive" title="b3">C3</font>, <font color="Olive" title="b4">A1</font>, <font color="Olive" title="b5">A2</font>, <font color="Olive" title="b6">A3</font>, <font color="Olive" title="b7">B1</font>, <font color="Olive" title="b8">B2</font>, <font color="Olive" title="b9">B3</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">a1</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">C1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</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">C2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</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">C3</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">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c11">t</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">A1</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">a3</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c4">a1</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">A2</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">a2</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c4">a1</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">A3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Olive" title="b7">B1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c7">b3</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">B2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c5">b2</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">B3</font> &amp; <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b2">C2</font>,<font color="Olive" title="b3">C3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> holds <br/> ex <font color="Olive" title="b10">O</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c11">t</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">O</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">C1</font>, <font color="Maroon" title="c13">C2</font>, <font color="Maroon" title="c14">C3</font>, <font color="Maroon" title="c15">A1</font>, <font color="Maroon" title="c16">A2</font>, <font color="Maroon" title="c17">A3</font>, <font color="Maroon" title="c18">B1</font>, <font color="Maroon" title="c19">B2</font>, <font color="Maroon" title="c20">B3</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"> ( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">a1</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">C1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</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">C2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</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">C3</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">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c11">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">A1</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">a3</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c4">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A2</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">a2</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c4">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">A3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">B1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c7">b3</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="c19">B2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c5">b2</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="c20">B3</font> &amp; <font color="Maroon" title="c12">C1</font>,<font color="Maroon" title="c13">C2</font>,<font color="Maroon" title="c14">C3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> implies  ex <font color="Olive" title="b1">O</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c11">t</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">O</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">a1</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">C1</font>
 <span class="kw">and </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c5">b2</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">C2</font>
 <span class="kw">and </span><br/><a NAME="E4:26"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">a3</font>,<font color="Maroon" title="c7">b3</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">C3</font>
 <span class="kw">and </span><br/><a NAME="E5:26"/><span class="lab"><font color="Green" title="E21">A5</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">a3</font>,<font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c11">t</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">A1</font>
 <span class="kw">and </span><br/><a NAME="E6:26"/><span class="lab"><font color="Green" title="E22">A6</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">a3</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c4">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">A2</font>
 <span class="kw">and </span><br/><a NAME="E7:26"/><span class="lab"><font color="Green" title="E23">A7</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">a2</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c4">a1</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">A3</font>
 <span class="kw">and </span><br/><a NAME="E8:26"/><span class="lab"><font color="Green" title="E24">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="c11">t</font>,<font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c7">b3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">B1</font>
 <span class="kw">and </span><br/><a NAME="E9:26"/><span class="lab"><font color="Green" title="E25">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="c3">b1</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c7">b3</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="c19">B2</font>
 <span class="kw">and </span><br/><a NAME="E10:26"/><span class="lab"><font color="Green" title="E26">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="c3">b1</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c5">b2</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="c20">B3</font>
 <span class="kw">and </span><br/><a NAME="E11:26"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c12">C1</font>,<font color="Maroon" title="c13">C2</font>,<font color="Maroon" title="c14">C3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <span class="kw">and </span><br/><a NAME="E12:26"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a3</font> )
 <span class="kw">and </span><br/><a NAME="E13:26"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font>
 <span class="kw">and </span><br/><a NAME="E14:26"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font>
 <span class="kw">and </span><br/><a NAME="E15:26"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font>
 <span class="kw">and </span><br/><a NAME="E16:26"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b2</font> &amp; <font color="Maroon" title="c8">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c11">t</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">O</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">o9</font> = <font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c22">b19</font> = <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c23">a19</font> = <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c24">b29</font> = <font color="Maroon" title="c5">b2</font>, <font color="Maroon" title="c25">a29</font> = <font color="Maroon" title="c6">a2</font>, <font color="Maroon" title="c26">b39</font> = <font color="Maroon" title="c7">b3</font>, <font color="Maroon" title="c27">a39</font> = <font color="Maroon" title="c8">a3</font>, <font color="Maroon" title="c28">r9</font> = <font color="Maroon" title="c9">r</font>, <font color="Maroon" title="c29">s9</font> = <font color="Maroon" title="c10">s</font>, <font color="Maroon" title="c30">t9</font> = <font color="Maroon" title="c11">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CPS</font> ;<br/>
<a NAME="E18:26"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C2</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E19:26"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c10">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E26">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="E20:26"/>
( <font color="Maroon" title="c5">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B3</font> &amp; <font color="Maroon" title="c3">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c20">B3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E26">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="E21:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c24">b29</font>,<font color="Maroon" title="c29">s9</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="#E19:26"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E22:26"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E25">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="E23:26"/>
( <font color="Maroon" title="c7">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B2</font> &amp; <font color="Maroon" title="c3">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c19">B2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E25">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="E24:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A21</font></span>: 
<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c26">b39</font>,<font color="Maroon" title="c28">r9</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="#E22:26"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E25:26"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c11">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E24">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="E26:26"/>
( <font color="Maroon" title="c7">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">B1</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">B1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E24">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="E27:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<font color="Maroon" title="c24">b29</font>,<font color="Maroon" title="c26">b39</font>,<font color="Maroon" title="c30">t9</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="#E25:26"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E28:26"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<font color="Maroon" title="c10">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E29:26"/>
( <font color="Maroon" title="c6">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A3</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">A3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E30:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A25</font></span>: 
<font color="Maroon" title="c23">a19</font>,<font color="Maroon" title="c25">a29</font>,<font color="Maroon" title="c29">s9</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="#E28:26"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E31:26"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C3</font> &amp; <font color="Maroon" title="c7">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E32:26"/>
<font color="Maroon" title="c8">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E33:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A27</font></span>: 
<font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c26">b39</font>,<font color="Maroon" title="c27">a39</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="#E31:26"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E34:26"/>
<font color="Maroon" title="c6">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E35:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A28</font></span>: 
<font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c24">b29</font>,<font color="Maroon" title="c25">a29</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="#E18:26"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E36:26"/><span class="lab"><font color="Green" title="E45">A29</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E37:26"/>
( <font color="Maroon" title="c8">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A2</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E38:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A30</font></span>: 
<font color="Maroon" title="c23">a19</font>,<font color="Maroon" title="c27">a39</font>,<font color="Maroon" title="c28">r9</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="#E36:26"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E39:26"/><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<font color="Maroon" title="c11">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E40:26"/>
( <font color="Maroon" title="c8">a3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">A1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">A1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E41:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<font color="Maroon" title="c25">a29</font>,<font color="Maroon" title="c27">a39</font>,<font color="Maroon" title="c30">t9</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="#E39:26"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E42:26"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C1</font> &amp; <font color="Maroon" title="c3">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E43:26"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
( not <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c24">b29</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c26">b39</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c24">b29</font>,<font color="Maroon" title="c26">b39</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E51">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1"/>
<font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c26">b39</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">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><span class="kw"> such that </span><br/><a NAME="E3:26_1_1"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c3">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c7">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">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/><a NAME="E4:26_1_1"/>
( <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">C1</font> &amp; <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="txt" href="#E31:26"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:26_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</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="E52">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1_2"/>
<font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c24">b29</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">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><span class="kw"> such that </span><br/><a NAME="E3:26_1_2"/><span class="lab"><font color="Green" title="E53">A38</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c3">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">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/><a NAME="E4:26_1_2"/>
( <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">C1</font> &amp; <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:26_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</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="E53">A39</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1_3"/>
<font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c24">b29</font>,<font color="Maroon" title="c26">b39</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">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><span class="kw"> such that </span><br/><a NAME="E3:26_1_3"/><span class="lab"><font color="Green" title="E54">A40</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">K</font> &amp; <font color="Maroon" title="c7">b3</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">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/><a NAME="E4:26_1_3"/>
( <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C2</font> &amp; <font color="Maroon" title="c31">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">C3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E31:26"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E3:26_1_3"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:26_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E4:26_1"/>
( <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c24">b29</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c26">b39</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c24">b29</font>,<font color="Maroon" title="c26">b39</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"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E5:26_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E53">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E44:26"/>
<font color="Maroon" title="c4">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<a NAME="E45:26"/><span class="kw">then </span>
<font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c22">b19</font>,<font color="Maroon" title="c23">a19</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="#E42:26"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<a NAME="E46:26"/><span class="kw">then </span>
<font color="Maroon" title="c30">t9</font>,<font color="Maroon" title="c28">r9</font>,<font color="Maroon" title="c29">s9</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="#E1:26"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E30:26"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E38:26"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="txt" href="#E27:26"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E41:26"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="txt" href="#E35:26"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E43:26"><span class="lab"><font color="Green" title="E50">A34</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">O</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><span class="kw"> such that </span><br/><a NAME="E48:26"/><span class="lab"><font color="Green" title="E51">A41</font></span>: 
( <font color="Maroon" title="c11">t</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">O</font> &amp; <font color="Maroon" title="c9">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">O</font> &amp; <font color="Maroon" title="c10">s</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c31">O</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/>
<a NAME="E49:26"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c11">t</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="c31">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:26"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E50:26"/>
 ex <font color="Olive" title="b1">O</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 <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c11">t</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">O</font>
 ; <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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