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

<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>











<b>assume </b><a NAME="E1:4"/><i><font color="Green">E23</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font> )
 ;<br/>

<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:4_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E3:4_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font>
 ;<br/><a NAME="E2:4_2_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>hence </b><a NAME="E3:4_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/></div><b>end;</b></div><a NAME="E3:4_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E4:4_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_3_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font>
 ;<br/><a NAME="E2:4_3_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E3:4_3_1"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:4_3_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><b>hence </b><a NAME="E5:4_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:4_3"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E4:4_3"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:4"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 <b>by </b><i>, , , </i>;<br/>


</div>
