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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="Red">::  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">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b3">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="b3">P</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>  ex <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 <br/>( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b2">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="Olive" title="b2">P</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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 <br/>( <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> &amp; not <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P</font> )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">B</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="E2:103"/><span class="lab"><font color="Green" title="E56">A1</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">and </span><br/><a NAME="E3:103"/><span class="lab"><font color="Green" title="E57">A2</font></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="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="incsp_1.html#E60"><span class="lab"><font color="Green" title="E36">Lm1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">C</font>, <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="E5:103"/><span class="lab"><font color="Green" title="E58">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="#E2:103"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="incsp_1.html#T77" target="_self" title="INCSP_1:th.77">Th77</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c7">P</font> =  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c7">P</font> &amp; not <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c7">P</font> )</span><br/>

<a NAME="E6:103"/><span class="lab"><font color="Green" title="E59">A4</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="c5">C</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c4">B</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="#E5:103"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="enumset1.html#T105" title="ENUMSET1:th.105">ENUMSET1:105</a></span>;<br/>
<a NAME="E7:103"/><span class="kw">then </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="c6">D</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#T38" target="_self" title="INCSP_1:th.38">Th38</a></span>;<br/>
<a NAME="E8:103"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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="c5">C</font>,<font color="Maroon" title="c6">D</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="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:103"/>
<font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c7">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">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c7">P</font></span><br/>

<a NAME="E10:103"/>
( <font color="Maroon" title="c4">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c7">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="c5">C</font>,<font color="Maroon" title="c6">D</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="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="c7">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:103"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></span>;<br/>
<a NAME="E11:103"/><span class="kw">then </span>
( <font color="Maroon" title="c4">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c7">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="c5">C</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c4">B</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="c7">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:103"/>
not <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c7">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E6:103"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="incsp_1.html#D7" target="_self" title="INCSP_1:def.7">Def7</a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</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/>


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