<?xml version="1.0"?>
<div class="add">

<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>, <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#K4" title="INCSP_1:func.4">Plane</a> <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 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="b3">C</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="Red">::  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#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">P2</font> =  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">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="c3">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">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="c4">C</font>;<br/>
<span class="kw">assume </span><a NAME="E1:92"/><span class="lab"><font color="Green" title="E49">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="Red">::  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#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span></span><br/>

<a NAME="E2:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A2</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="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T36" target="_self" title="INCSP_1:th.36">Th36</a></span>;<br/>
<a NAME="E3:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A3</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="E4:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A4</font></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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></span>;<br/>
<a NAME="E5:92"/>
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">C</font>,<font color="Maroon" title="c3">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="#E1:92"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="enumset1.html#T98" title="ENUMSET1:th.98">ENUMSET1:98</a></span>;<br/>
<a NAME="E6:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A5</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">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T36" target="_self" title="INCSP_1:th.36">Th36</a></span>;<br/>
<a NAME="E7:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A6</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">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>  <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">C</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="E8:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A7</font></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="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></span>;<br/>
<a NAME="E9:92"/>
<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">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>  <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">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></span>;<br/>
<a NAME="E10:92"/><span class="kw">then </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="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></span>;<br/>
<a NAME="E11:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A8</font></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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="incsp_1.html#T65" target="_self" title="INCSP_1:th.65">Th65</a></span>;<br/>
<a NAME="E12:92"/><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="c4">C</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> <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="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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="incsp_1.html#D22" target="_self" title="INCSP_1:def.22">Def22</a></span>;<br/>
<a NAME="E13:92"/><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">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>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:92"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="incsp_1.html#T35" target="_self" title="INCSP_1:th.35">Th35</a></span>;<br/>
<a NAME="E14:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<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#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></span>;<br/>
<a NAME="E15:92"/>
 <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#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E11:92"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="ref" href="incsp_1.html#D22" target="_self" title="INCSP_1:def.22">Def22</a></span>;<br/>
<a NAME="E16:92"/><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="incsp_1.html#R5" title="INCSP_1:pred.5">on</a>  <a href="incsp_1.html#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="incsp_1.html#T35" target="_self" title="INCSP_1:th.35">Th35</a></span>;<br/>
<a NAME="E17:92"/><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#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:92"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="incsp_1.html#T19" target="_self" title="INCSP_1:th.19">Th19</a></span>;<br/>
<a NAME="E18:92"/><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#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T43" title="ENUMSET1:th.43">ENUMSET1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:92"/>
 <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#K4" title="INCSP_1:func.4">Plane</a> <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="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="c4">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E49">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
