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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"> 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/>

<span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:21"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E7:21"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<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="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E8:21"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c4">X</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="#E4:21"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font>;<br/>
<a NAME="E9:21"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</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="#E8:21"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="aff_4.html#T27" title="AFF_4:th.27">AFF_4:27</a></span>;<br/>
<a NAME="E10:21"/>
<font color="Maroon" title="c4">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</a></span>;<br/>
<a NAME="E11:21"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></span>;<br/>
<a NAME="E12:21"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="afproj.html#T9" target="_self" title="AFPROJ:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:21"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</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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