<?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="firebrick">::  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>  ex <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">L1</font>, <font color="Olive" title="b4">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 <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">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="Olive" title="b3">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="Olive" title="b4">L2</font> &amp; (  for <font color="Olive" title="b5">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="b2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b5">P</font> or  not <font color="Olive" title="b3">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b5">P</font> or  not <font color="Olive" title="b4">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b5">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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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 <br/>( <font color="Maroon" title="c2">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="Maroon" title="c2">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="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L2</font> &amp; (  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> ) ) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E2:72"/><span class="lab"><font color="Green" title="E55">A1</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="c2">A</font>,<font color="Maroon" title="c2">A</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="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D12" target="_self" title="INCSP_1:def.12">Def12</a></span>;<br/>
<a NAME="E3:72"/>
<font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="incsp_1.html#T4" target="_self" title="INCSP_1:th.4">Th14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">L</font>, <font color="Maroon" title="c5">L1</font>, <font color="Maroon" title="c6">L2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E5:72"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c5">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">L2</font>
 <span class="kw">and </span><br/><a NAME="E6:72"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c5">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E7:72"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c6">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E8:72"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
 not <font color="Maroon" title="c4">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E9:72"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">and </span><br/><a NAME="E10:72"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font>
 <span class="kw">and </span><br/><a NAME="E11:72"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T50" target="_self" title="INCSP_1:th.50">Th80</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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="E13:72"/><span class="lab"><font color="Green" title="E63">A9</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="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E14:72"/><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="incsp_1.html#E39"><span class="lab"><font color="Green" title="E36">Lm1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">C</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="E16:72"/><span class="lab"><font color="Green" title="E65">A11</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="c8">C</font>
 <span class="kw">and </span><br/><a NAME="E17:72"/><span class="lab"><font color="Green" title="E66">A12</font></span>: 
<font color="Maroon" title="c8">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="incsp_1.html#E39"><span class="lab"><font color="Green" title="E36">Lm1</font></span></a></span>;<br/>
<a NAME="E18:72"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
<font color="Maroon" title="c8">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E17:72"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/>
<a NAME="E19:72"/><span class="lab"><font color="Green" title="E68">A14</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="c7">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="c5">L1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E14:72"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<a NAME="E20:72"/><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="c7">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="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th35</a></span>;<br/>
<a NAME="E21:72"/><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="c7">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="c8">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> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:72"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/>
<a NAME="E22:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A15</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="c7">B</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c3">P</font>
 
<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">take </span>
<font color="Maroon" title="c4">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">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 <br/>( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L2</font> &amp; (  for <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>  holds <br/>(  not <font color="Maroon" title="c4">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b3">P</font> or  not <font color="Olive" title="b1">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b3">P</font> or  not <font color="Olive" title="b2">L2</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">take </span>
<font color="Maroon" title="c5">L1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L2</font> &amp; (  for <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>  holds <br/>(  not <font color="Maroon" title="c4">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> or  not <font color="Maroon" title="c5">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">P</font> or  not <font color="Olive" title="b1">L2</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">take </span>
<font color="Maroon" title="c6">L2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font> &amp; (  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="c4">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="c5">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="c6">L2</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">thus </span><a NAME="E23:72"/>
( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c5">L1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:72"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E62">A8</font></span></a></span>; <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">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="c4">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="c5">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="c6">L2</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">given </span><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><span class="kw"> such that </span><a NAME="E25:72"/><span class="lab"><font color="Green" title="E70">A16</font></span>: 
<font color="Maroon" title="c4">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">and </span><br/><a NAME="E26:72"/><span class="lab"><font color="Green" title="E71">A17</font></span>: 
<font color="Maroon" title="c5">L1</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">and </span><br/><a NAME="E27:72"/><span class="lab"><font color="Green" title="E72">A18</font></span>: 
<font color="Maroon" title="c6">L2</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c9">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E28:72"/><span class="lab"><font color="Green" title="E73">A19</font></span>: 
<font color="Maroon" title="c8">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c9">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:72"><span class="lab"><font color="Green" title="E72">A18</font></span></a>, <a class="txt" href="#E17:72"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/>
<a NAME="E29:72"/><span class="lab"><font color="Green" title="E74">A20</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="c8">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="c6">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="txt" href="#E17:72"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c10">K</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><a NAME="E2:72_1"/><span class="lab"><font color="Green" title="E75">A21</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="c7">B</font>,<font color="Maroon" title="c8">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="c10">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:72_1"/>
<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="c8">C</font>,<font color="Maroon" title="c7">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="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1"><span class="lab"><font color="Green" title="E75">A21</font></span></a>, <a class="ref" href="enumset1.html#T57" title="ENUMSET1:th.57">ENUMSET1:57</a></span>;<br/><a NAME="E4:72_1"/><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="c8">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="c7">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="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/><a NAME="E5:72_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A22</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="c8">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="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T8" target="_self" title="INCSP_1:th.8">Th18</a></span>;<br/><a NAME="E6:72_1"/>
<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="c7">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="c8">C</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="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1"><span class="lab"><font color="Green" title="E75">A21</font></span></a>, <a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a></span>;<br/><a NAME="E7:72_1"/><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="c7">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="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#T8" target="_self" title="INCSP_1:th.8">Th18</a></span>;<br/><a NAME="E8:72_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:72"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="txt" href="#E19:72"><span class="lab"><font color="Green" title="E68">A14</font></span></a>, <a class="ref" href="incsp_1.html#D10" target="_self" title="INCSP_1:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E9:72_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E16:72"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="txt" href="#E29:72"><span class="lab"><font color="Green" title="E74">A20</font></span></a>, <a class="txt" href="#E5:72_1"><span class="lab"><font color="Green" title="E76">A22</font></span></a>, <a class="ref" href="incsp_1.html#D10" target="_self" title="INCSP_1:def.10">Def10</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/></div><span class="kw">end;</span></div>
<a NAME="E31:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A23</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="c7">B</font>,<font color="Maroon" title="c8">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incsp_1.html#D6" target="_self" title="INCSP_1:def.6">Def6</a></span>;<br/>
<a NAME="E32:72"/>
<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="c7">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:72"><span class="lab"><font color="Green" title="E71">A17</font></span></a>, <a class="txt" href="#E19:72"><span class="lab"><font color="Green" title="E68">A14</font></span></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th35</a></span>;<br/>
<a NAME="E33:72"/><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="c7">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="c8">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> <font color="Maroon" title="c9">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:72"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="ref" href="incsp_1.html#T9" target="_self" title="INCSP_1:th.9">Th19</a></span>;<br/>
<a NAME="E34:72"/><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="c7">B</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c9">Q</font>
 
<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="E35:72"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E25:72"><span class="lab"><font color="Green" title="E70">A16</font></span></a>, <a class="txt" href="#E22:72"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E31:72"><span class="lab"><font color="Green" title="E75">A23</font></span></a>, <a class="ref" href="incsp_1.html#D13" target="_self" title="INCSP_1:def.13">Def13</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/>


</div>
