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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="incsp_1.html#NM4" title="INCSP_1:NM.4">IncSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b2">P</font> holds <br/> ex <font color="Olive" title="b3">L</font>, <font color="Olive" title="b4">L1</font>, <font color="Olive" title="b5">L2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">L2</font> &amp; <font color="Olive" title="b4">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b5">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> &amp;  not <font color="Olive" title="b3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L</font> &amp; <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">L1</font> &amp; <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b5">L2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  st <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> holds <br/> ex <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">L1</font>, <font color="Olive" title="b4">L2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b3">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">L2</font> &amp; <font color="Olive" title="b3">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b4">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> &amp;  not <font color="Olive" title="b2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b4">L2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">P</font> implies  ex <font color="Olive" title="b1">L</font>, <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">L2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Olive" title="b2">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b3">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp;  not <font color="Olive" title="b1">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L2</font> ) )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E3:71"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
 not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is  <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T46" target="_self" title="INCSP_1:th.46">Th76</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">D</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E5:71"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is  <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="incsp_1.html#T45" target="_self" title="INCSP_1:th.45">Th75</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:71"/>
<font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">L</font>, <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">L2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Olive" title="b2">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b3">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp;  not <font color="Olive" title="b1">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L2</font> )</span><br/>

<a NAME="E7:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/>
<a NAME="E8:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">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="c2">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T2" title="ENUMSET1:th.2">ENUMSET1:2</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c7">L3</font> =  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">L2</font> &amp; <font color="Olive" title="b1">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp;  not <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L2</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c8">L1</font> =  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">L2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c8">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">L2</font> &amp; <font color="Maroon" title="c8">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b1">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp;  not <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L2</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c9">L2</font> =  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">L2</font> &amp; <font color="Maroon" title="c8">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp;  not <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> )</span><br/>

<a NAME="E9:71"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th36</a></span>;<br/>
<a NAME="E10:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">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="c8">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></span>;<br/>
<a NAME="E11:71"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
 not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is  <a href="incsp_1.html#V3" title="INCSP_1:attr.3">linear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="enumset1.html#T57" title="ENUMSET1:th.57">ENUMSET1:57</a></span>;<br/>
<a NAME="E12:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th36</a></span>;<br/>
<a NAME="E13:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</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="c9">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></span>;<br/>
<a NAME="E14:71"/><span class="kw">then </span>
( <font color="Maroon" title="c4">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> implies <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c9">L2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T8" target="_self" title="INCSP_1:th.8">Th18</a></span>;<br/>
<a NAME="E15:71"/><span class="kw">then </span>
( <font color="Maroon" title="c4">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> implies <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c4">B</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">L2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:71"/>
<font color="Maroon" title="c8">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp;  not <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> )</span><br/>

<a NAME="E17:71"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/>
<a NAME="E18:71"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:71"/>
<font color="Maroon" title="c8">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:71"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp;  not <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> )</span><br/>

<a NAME="E20:71"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="enumset1.html#T2" title="ENUMSET1:th.2">ENUMSET1:2</a></span>;<br/>
<a NAME="E21:71"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/>
<a NAME="E22:71"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:71"/>
<font color="Maroon" title="c9">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:71"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="txt" href="#E13:71"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> )</span><br/>

<a NAME="E24:71"/>
 not <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> is  <a href="incsp_1.html#V4" title="INCSP_1:attr.4">planar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="enumset1.html#T63" title="ENUMSET1:th.63">ENUMSET1:63</a></span>;<br/>
<a NAME="E25:71"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T16" target="_self" title="INCSP_1:th.16">Th37</a></span>;<br/>
<a NAME="E26:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D</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="c7">L3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></span>;<br/>
<a NAME="E27:71"/><span class="kw">then </span>
( <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> implies <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th35</a></span>;<br/>
<a NAME="E28:71"/><span class="kw">then </span>
( <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> implies <font color="Maroon" title="c6">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T3" target="_self" title="INCSP_1:th.3">Th13</a></span>;<br/>
<a NAME="E29:71"/><span class="kw">then </span>
( <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> implies <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/>
<a NAME="E30:71"/><span class="kw">then </span>
( <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font> implies <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T6" title="ENUMSET1:th.6">ENUMSET1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:71"/>
 not <font color="Maroon" title="c7">L3</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E32:71"/>
( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">L3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:71"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E26:71"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
