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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <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 ( for <font color="Olive" title="b4">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b1">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b4">P</font> or  not <font color="Olive" title="b2">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b4">P</font> or  not <font color="Olive" title="b3">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b4">P</font> ) ) &amp;  ex <font color="Olive" title="b4">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b4">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="Olive" title="b4">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="Olive" title="b4">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L2</font> ) holds <br/><font color="Olive" title="b1">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">L1</font></span><br/><b>let </b><font color="Maroon" title="c2">L</font>, <font color="Maroon" title="c3">L1</font>, <font color="Maroon" title="c4">L2</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Maroon" title="c2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> or  not <font color="Maroon" title="c3">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> or  not <font color="Maroon" title="c4">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> ) ) &amp;  ex <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">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="Maroon" title="c3">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="Maroon" title="c4">L2</font> ) implies <font color="Maroon" title="c2">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L1</font> )</span><br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E37">A1</font></i>: 
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>  holds <br/>(  not <font color="Maroon" title="c2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> or  not <font color="Maroon" title="c3">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> or  not <font color="Maroon" title="c4">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( for <font color="Olive" title="b1">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">L</font> or  not <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L1</font> or  not <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L2</font> ) or <font color="Maroon" title="c2">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L1</font> )</span><br/>

<b>given </b><font color="Maroon" title="c5">A</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><a NAME="E3:39"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">L</font>
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L1</font>
 <b>and </b><br/><a NAME="E5:39"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L1</font></span><br/>

<b>assume </b><a NAME="E6:39"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c6">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E8:39"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E9:39"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E60"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">C</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E11:39"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">C</font>
 <b>and </b><br/><a NAME="E12:39"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L1</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E60"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">D</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E14:39"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">D</font>
 <b>and </b><br/><a NAME="E15:39"/><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c8">D</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E60"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/>
<a NAME="E16:39"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</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="c2">L</font>
 
<b>by </b><i><a class="txt" href="#E4:39"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E6:39"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E9:39"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E12:39"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</a></i>;<br/>
<a NAME="E17:39"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">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>
 
<b>by </b><i><a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a></i>;<br/>
<a NAME="E18:39"/><b>then </b>
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T38" target="_self" title="INCSP_1:th.38">Th38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">Q</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E20:39"/><i><font color="Green" title="E48">A12</font></i>: 
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">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="c9">Q</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a></i>;<br/>
<a NAME="E21:39"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">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="c9">Q</font>
 
<b>by </b><i><a class="txt" href="#E20:39"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></i>;<br/>
<a NAME="E22:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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="c9">Q</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">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="c2">L</font> )
 
<b>by </b><i><a class="txt" href="#E3:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E9:39"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a>, <a class="ref" href="incsp_1.html#T21" target="_self" title="INCSP_1:th.21">Th21</a></i>;<br/>
<a NAME="E23:39"/><b>then </b><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="#E8:39"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></i>;<br/>
<a NAME="E24:39"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c6">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="c3">L1</font>
 
<b>by </b><i><a class="txt" href="#E4:39"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E6:39"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E9:39"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E12:39"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</a></i>;<br/>
<a NAME="E25:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">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="c6">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="c3">L1</font> &amp; <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c7">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c9">Q</font> )
 
<b>by </b><i><a class="txt" href="#E20:39"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th15</a>, <a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></i>;<br/>
<a NAME="E26:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">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="c3">L1</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">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="c9">Q</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a>, <a class="ref" href="incsp_1.html#T20" target="_self" title="INCSP_1:th.20">Th20</a></i>;<br/>
<a NAME="E27:39"/><b>then </b><i><font color="Green" title="E50">A14</font></i>: 
<font color="Maroon" title="c3">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="#E11:39"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></i>;<br/>
<a NAME="E28:39"/>
( <font color="Maroon" title="c5">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c8">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c9">Q</font> )
 
<b>by </b><i><a class="txt" href="#E20:39"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="incsp_1.html#T15" target="_self" title="INCSP_1:th.15">Th15</a></i>;<br/>
<a NAME="E29:39"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c8">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="c4">L2</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c8">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="c9">Q</font> )
 
<b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="#E15:39"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></i>;<br/>
<a NAME="E30:39"/><b>then </b>
<font color="Maroon" title="c4">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="txt" href="#E14:39"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E31:39"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="#E23:39"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="#E27:39"><i><font color="Green" title="E50">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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