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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>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>  st  not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">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>  holds <br/> <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Olive" title="b3">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</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"> (  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="c3">B</font>,<font color="Maroon" title="c4">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>  implies  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:59"/><span class="lab"><font color="Green" title="E48">A1</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="c3">B</font>,<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>)</span><br/>

<a NAME="E2:59"/><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="c3">B</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="E3:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</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="c3">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>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</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="E4:59"/><span class="kw">then </span>
( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) &amp; <font color="Maroon" title="c3">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<a NAME="E5:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A3</font></span>: 
( <font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</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="c3">B</font>,<font color="Maroon" title="c4">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>  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T2" target="_self" title="INCSP_1:th.2">Th12</a></span>;<br/>
<a NAME="E6:59"/><span class="kw">then </span>
 <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E48">A1</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#D21" target="_self" title="INCSP_1:def.21">Def21</a></span>;<br/>
<a NAME="E7:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A4</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="c3">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>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th35</a></span>;<br/>
<a NAME="E8:59"/>
<font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E50">A3</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#D21" target="_self" title="INCSP_1:def.21">Def21</a></span>;<br/>
<a NAME="E9:59"/><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="c3">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="c4">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>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/>
<a NAME="E10:59"/><span class="kw">then </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="c3">B</font>,<font color="Maroon" title="c4">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#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>)
 
<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="E11:59"/>
 <a href="incsp_1.html#K2" title="INCSP_1:func.2">Plane</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="incsp_1.html#K3" title="INCSP_1:func.3">Plane</a> (<font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="incsp_1.html#D20" target="_self" title="INCSP_1:def.20">Def20</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/>


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