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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>;<br/><b>let </b><font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font>
 ;<br/>

<a NAME="E4:40"/><i><font color="Green" title="E40">A4</font></i>: 
( <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="c6">A</font> implies <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1"/>
<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="c6">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E41">A5</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c6">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="c7">p</font>,<font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">q</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#D4" target="_self" title="AFF_1:def.4">Def4</a></i>;<br/>
<a NAME="E4:40_1"/>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:40"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="aff_1.html#E3:40_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="aff_1.html#T36" target="_self" title="AFF_1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E5:40_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E3:40_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="aff_1.html#T14" target="_self" title="AFF_1:th.14">Th14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40"/>
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> implies <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="c6">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>
 ;<br/>

<a NAME="E2:40_2"/>
<font color="Maroon" title="c6">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">c</font>,<font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:40"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="aff_1.html#E2:40"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="aff_1.html#E3:40"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="aff_1.html#E37"><i><font color="Green" title="E25">Lm6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:40_2"/>
<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="c6">A</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E3:40"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="aff_1.html#E1:40_2"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="aff_1.html#D4" target="_self" title="AFF_1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:40"/>
( <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="c6">A</font> iff <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E4:40"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>


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