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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">K</font>, <font color="Olive" title="b2">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">L</font> &amp;  ex <font color="Olive" title="b3">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="b3">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L</font> ) holds <br/> <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Olive" title="b2">L</font>,<font color="Olive" title="b1">K</font></span><br/><b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">L</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"> ( <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L</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">K</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">L</font> ) implies  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> )</span><br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L</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">K</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">L</font> ) or  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> )</span><br/>

<b>given </b><font color="Maroon" title="c4">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="E3:57"/><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">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="E5:57"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E6:57"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</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="c6">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="E8:57"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">C</font>
 <b>and </b><br/><a NAME="E9:57"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">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>set </b><font color="Maroon" title="c7">P1</font> =  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font>;<br/>
<b>set </b><font color="Maroon" title="c8">P2</font> =  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font>;<br/>
<a NAME="E10:57"/>
( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E3:57"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="incsp_1.html#D22" target="_self" title="INCSP_1:def.22">Def22</a></i>;<br/>
<a NAME="E11:57"/><b>then </b>
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="#E3:57"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E6:57"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="#E9:57"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></i>;<br/>
<a NAME="E12:57"/><b>then </b><i><font color="Green" title="E54">A7</font></i>: 
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">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>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">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>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c9">L1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green" title="E55">A8</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">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="c9">L1</font>
 <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="E4:57_1"/>
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font> &amp; <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font> &amp; <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">L1</font> )
 <b>by </b><i><a class="txt" href="#E3:57_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self" title="INCSP_1:th.12">Th12</a></i>;<br/><a NAME="E5:57_1"/><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="c4">A</font>,<font color="Maroon" title="c5">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="c9">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="c4">A</font>,<font color="Maroon" title="c5">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">K</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">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">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="c4">A</font>,<font color="Maroon" title="c6">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">L</font> )
 <b>by </b><i><a class="txt" href="#E3:57"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E6:57"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="#E9:57"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/><a NAME="E6:57_1"/><b>then </b>
( <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L1</font> &amp; <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L1</font> )
 <b>by </b><i><a class="txt" href="#E5:57"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="#E8:57"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="incsp_1.html#D10" target="_self" title="INCSP_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E7:57_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:57"><i><font color="Green" title="E48">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:57"/>
 <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="#E12:57"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="incsp_1.html#D13" target="_self" title="INCSP_1:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
