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

<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">x</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b4">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">K</font> &amp; not <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">K</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b4">K</font> &amp; not <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> holds <br/> not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>; <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">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> &amp; not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> holds <br/> not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">K</font> &amp; not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E2:72"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E3:72"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> or  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:72"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E5:72"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:72"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="aff_1.html#T15" target="_self" title="AFF_1:th.15">Th15</a></span>;<br/>
<a NAME="E7:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></span>;<br/>
<a NAME="E8:72"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></span>;<br/>
<a NAME="E9:72"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="aff_1.html#D5" target="_self" title="AFF_1:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:72"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="aff_1.html#T59" target="_self" title="AFF_1:th.59">Th59</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>
