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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"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <font color="Maroon" title="c1">AS</font></span><br/>

<b>assume </b><a NAME="E1:21"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <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"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <font color="Maroon" title="c1">AS</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c3">X</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">X</font>;<br/>
<a NAME="E7:21"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">X</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="txt" href="#E5:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="aff_4.html#T27" title="AFF_4:th.27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="#E5:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/>
<a NAME="E9:21"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="#E5:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E7:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="afproj.html#T9" target="_self" title="AFPROJ:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E10:21"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</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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