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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font>, <font color="Olive" title="b5">p</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="b6">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">a</font>,<font color="Olive" title="b2">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b6">K</font> &amp; <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b6">K</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">b'</font> &amp; <font color="Olive" title="b5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">K</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">K</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">a'</font>, <font color="Maroon" title="c5">b'</font>, <font color="Maroon" title="c6">p</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:()"><i><font color="Red">::  thesis: </font></i></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">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> &amp; <font color="Maroon" title="c6">p</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="c2">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="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon" title="c4">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b'</font></span><br/><b>let </b><font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c7">K</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b'</font> )</span><br/>













<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E51">A1</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c7">K</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b'</font></span><br/>

<b>set </b><font color="Maroon" title="c8">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E2:59"/><i><font color="Green" title="E52">A2</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></i>;<br/>
<a NAME="E3:59"/><i><font color="Green" title="E53">A3</font></i>: 
( <font color="Maroon" title="c4">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="c6">p</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">b'</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="c6">p</font>,<font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></i>;<br/>
<b>assume </b><a NAME="E4:59"/><i><font color="Green" title="E54">A4</font></i>: 
<font color="Maroon" title="c4">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:59"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>
 
<b>by </b><i><a class="txt" href="#E2:59"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="#E3:59"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="aff_1.html#E37"><i><font color="Green" title="E25">Lm6</font></i></a></i>;<br/>
<a NAME="E6:59"/><b>then </b><i><font color="Green" title="E55">A5</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">K</font>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E4:59"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="aff_1.html#D5" target="_self" title="AFF_1:def.5">Def5</a></i>;<br/>
<a NAME="E7:59"/>
<font color="Maroon" title="c6">p</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="c6">p</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></i>;<br/>
<a NAME="E8:59"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K</font>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E6:59"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="aff_1.html#T59" target="_self" title="AFF_1:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E9:59"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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