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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">a'</font>, <font color="Olive" title="b3">d</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="b4">A</font>, <font color="Olive" title="b5">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="b4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b5">K</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">A</font> &amp; <font color="Olive" title="b2">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">A</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">K</font> holds <br/> ex <font color="Olive" title="b6">d'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b6">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">K</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b6">d'</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">a'</font>, <font color="Maroon" title="c4">d</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>, <font color="Olive" title="b2">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">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">K</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c3">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> holds <br/> ex <font color="Olive" title="b3">d'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b3">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Olive" title="b3">d'</font> )</span><br/><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c3">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> implies  ex <font color="Olive" title="b1">d'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Olive" title="b1">d'</font> ) )</span><br/>











<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c3">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Olive" title="b1">d'</font> )</span><br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c5">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 
<b>by </b><i><a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Olive" title="b1">d'</font> )</span><br/><b>consider </b><font color="Maroon" title="c7">d'</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="E3:8_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c7">d'</font>
 <b>and </b><br/><a NAME="E4:8_1"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c7">d'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></i>;<br/><a NAME="E5:8_1"/>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c7">d'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>
 <b>by </b><i><a class="txt" href="#E3:8_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E6:8_1"/><b>then </b>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c7">d'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E1:8_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></i>;<br/><a NAME="E7:8_1"/><b>then </b>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c7">d'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">K</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self" title="AFF_4:th.3">Th3</a></i>;<br/><a NAME="E8:8_1"/><b>then </b>
<font color="Maroon" title="c7">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E9:8_1"/>
 ex <font color="Olive" title="b1">d'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Olive" title="b1">d'</font> )
 <b>by </b><i><a class="txt" href="#E4:8_1"><i><font color="Green" title="E12">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
 ex <font color="Olive" title="b1">d'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Olive" title="b1">d'</font> )
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</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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