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<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">a9</font>, <font color="Olive" title="b2">b9</font>, <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">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="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">a9</font>,<font color="Olive" title="b2">b9</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="b3">p</font>,<font color="Olive" title="b4">a</font>,<font color="Olive" title="b1">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b5">b</font>,<font color="Olive" title="b2">b9</font> &amp; <font color="Olive" title="b3">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="b4">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="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">b</font> holds <br/><font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a9</font>, <font color="Maroon" title="c3">b9</font>, <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">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">a9</font>,<font color="Maroon" title="c3">b9</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="c4">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">b9</font> &amp; <font color="Maroon" title="c4">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="c5">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="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b9</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a9</font>,<font color="Maroon" title="c3">b9</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="c4">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">b9</font> &amp; <font color="Maroon" title="c4">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="c5">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="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b9</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c2">a9</font>,<font color="Maroon" title="c3">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c7">K</font>
 <span class="kw">and </span><br/><a NAME="E2:73"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">a9</font>
 <span class="kw">and </span><br/><a NAME="E3:73"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c3">b9</font>
 <span class="kw">and </span><br/><a NAME="E4:73"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font>
 <span class="kw">and </span><br/><a NAME="E5:73"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font>
 <span class="kw">and </span><br/><a NAME="E6:73"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</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">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b9</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">a</font>;<br/>
<a NAME="E7:73"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
<font color="Maroon" title="c3">b9</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="c4">p</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></span>;<br/>
<a NAME="E8:73"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c4">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="c4">p</font>,<font color="Maroon" title="c5">a</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="E9:73"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c2">a9</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="c4">p</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:73"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c2">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b9</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="E11:73"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">a</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></span>;<br/>
<a NAME="E12:73"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">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="c2">a9</font>,<font color="Maroon" title="c3">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E10:73"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="aff_1.html#E37"><span class="lab"><font color="Green" title="E25">Lm6</font></span></a></span>;<br/>
<a NAME="E13:73"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">a</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E10:73"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="aff_1.html#D5" target="_self" title="AFF_1:def.5">Def5</a></span>;<br/>
<a NAME="E14:73"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="aff_1.html#T59" target="_self" title="AFF_1:th.59">Th59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:73"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</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/>


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