<?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">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/> 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/> for <font color="Olive" title="b4">Q</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/> ( ( <font color="Olive" title="b1">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Olive" title="b2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b4">Q</font> ) iff <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  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/> for <font color="Olive" title="b2">Q</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/> ( ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> ) iff <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 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> )
 ; <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">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/> for <font color="Olive" title="b2">Q</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/> ( ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> ) iff <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> )</span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:45"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">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="E7:45"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">and </span><br/><a NAME="E8:45"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c5">C</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#E39"><span class="lab"><font color="Green" title="E36">Lm1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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="E10:45"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">B</font>
 <span class="kw">and </span><br/><a NAME="E11:45"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</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="c7">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="E13:45"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<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="c6">B</font>,<font color="Maroon" title="c5">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="c7">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="E14:45"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
<font color="Maroon" title="c4">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="txt" href="#E13:45"><span class="lab"><font color="Green" title="E48">A9</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">take </span>
<font color="Maroon" title="c7">P</font>
; <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">Q</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/> ( ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">Q</font> &amp; <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">Q</font> ) iff <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">Q</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</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"> ( ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> ) iff <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font> )</span><br/>

<span class="kw">thus </span><a NAME="E15:45"/>
( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> implies <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</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="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font> implies ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:45_1"/>
<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">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<a NAME="E2:45_1"/><span class="kw">then </span>
 not <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">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="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="incsp_1.html#D10" target="_self" title="INCSP_1:def.10">Def10</a></span>;<br/>
<a NAME="E3:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A11</font></span>: 
 not <font color="Maroon" title="c5">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:45_1"/><span class="lab"><font color="Green" title="E51">A12</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:45_1"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
<font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</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="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font></span><br/>

<a NAME="E6:45_1"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>;<br/>
<a NAME="E7:45_1"/>
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c6">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E51">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="E8:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">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="c4">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c5">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="c8">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_1"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="incsp_1.html#T4" target="_self" title="INCSP_1:th.4">Th14</a></span>;<br/>
<a NAME="E9:45_1"/>
<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">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<a NAME="E10:45_1"/><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="c4">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c5">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="txt" href="#E10:45"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="incsp_1.html#T18" target="_self" title="INCSP_1:th.18">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:45_1"/>
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:45"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E8:45_1"><span class="lab"><font color="Green" title="E54">A15</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><span class="kw">end;</span></div>

<a NAME="E16:45"/>
<font color="Maroon" title="c6">B</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="txt" href="#E13:45"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="incsp_1.html#T4" target="_self" title="INCSP_1:th.4">Th14</a></span>;<br/>
<a NAME="E17:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<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">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="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="incsp_1.html#T3" target="_self" title="INCSP_1:th.3">Th13</a></span>;<br/>
<a NAME="E18:45"/>
<font color="Maroon" title="c5">C</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="txt" href="#E13:45"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="incsp_1.html#T4" target="_self" title="INCSP_1:th.4">Th14</a></span>;<br/>
<a NAME="E19:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A17</font></span>: 
<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">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> <font color="Maroon" title="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="incsp_1.html#T3" target="_self" title="INCSP_1:th.3">Th13</a></span>;<br/>
<a NAME="E20:45"/><span class="lab"><font color="Green" title="E52">A18</font></span>: 
<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">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<a NAME="E21:45"/>
<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">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="incsp_1.html#T1" target="_self" title="INCSP_1:th.1">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:45"/>
( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font> implies ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E20:45"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E19:45"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</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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