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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">q</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="b3">A</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="b3">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/><font color="Olive" title="b1">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b3">A</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">q</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">A</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> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/><font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b1">A</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">A</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="c4">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  implies <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 ; <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> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font></span>)</span></span><br/>

<a NAME="E2:41"/><b>then </b>
<font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 
<b>by </b><i><a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></i>;<br/>
<a NAME="E3:41"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:41"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>;<br/>
<a NAME="E4:41"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E5:41"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:41"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E6:41"/>
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:41"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</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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