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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"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</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="b5">M</font>, <font color="Olive" title="b6">N</font>, <font color="Olive" title="b7">P</font>, <font color="Olive" title="b8">Q</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">M</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">M</font> &amp; <font color="Olive" title="b3">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">N</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">N</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">P</font> &amp; <font color="Olive" title="b3">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">P</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">Q</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">Q</font> &amp; <font color="Olive" title="b5">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">N</font> &amp; <font color="Olive" title="b5">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b6">N</font> &amp; <font color="Olive" title="b7">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b8">Q</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; not <font color="Olive" title="b7">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b8">Q</font> holds <br/> ex <font color="Olive" title="b9">q</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="b9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">P</font> &amp; <font color="Olive" title="b9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">Q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">a9</font>, <font color="Maroon" title="c5">b9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font>, <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</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="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Maroon" title="c5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b1">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b4">Q</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; not <font color="Olive" title="b3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b4">Q</font> holds <br/> ex <font color="Olive" title="b5">q</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="b5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">M</font>, <font color="Maroon" title="c7">N</font>, <font color="Maroon" title="c8">P</font>, <font color="Maroon" title="c9">Q</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font> &amp; <font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c6">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c8">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c9">Q</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; not <font color="Maroon" title="c8">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">Q</font> implies  ex <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> ) )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">and </span><br/><a NAME="E2:22"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">and </span><br/><a NAME="E3:22"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">and </span><br/><a NAME="E4:22"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">and </span><br/><a NAME="E5:22"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">and </span><br/><a NAME="E6:22"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">and </span><br/><a NAME="E7:22"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">and </span><br/><a NAME="E8:22"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">and </span><br/><a NAME="E9:22"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">and </span><br/><a NAME="E10:22"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">and </span><br/><a NAME="E11:22"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<font color="Maroon" title="c8">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E12:22"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c9">Q</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">Q</font> or  ex <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> ) )</span><br/>

<a NAME="E13:22"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E14:22"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c7">N</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="#E10:22"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<a NAME="E15:22"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E16:22"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon" title="c6">M</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="#E10:22"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">Q</font> or  ex <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c10">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">and </span><br/><a NAME="E4:22_1"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">c</font>;<br/><a NAME="E5:22_1"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E6:22_1"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E7:22_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E16:22"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a>, <a class="ref" href="aff_1.html#T66" title="AFF_1:th.66">AFF_1:66</a></span>;<br/><a NAME="E8:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c10">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/><a NAME="E9:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A22</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/><a NAME="E10:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E11:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A24</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">c</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="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:22_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> )</span><br/><a NAME="E2:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A25</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E15:22"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E11:22_1"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E3:22_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c4">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E9:22_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E5:22_1_1"/><span class="lab"><font color="Green" title="E43">A26</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E6:22_1_1"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="aff_4.html#T1" target="_self" title="AFF_4:th.1">Th1</a></span>;<br/><a NAME="E7:22_1_1"/>
<font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E8:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E6:22_1_1"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E9:22_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E10:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A28</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E11:22_1_1"/>
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E15:22"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E12:22_1_1"/>
 ex <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_1_1"><span class="lab"><font color="Green" title="E45">A28</font></span></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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:22_1"/>
( <font color="Maroon" title="c8">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">Q</font> or  ex <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E10:22_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:22"/>
( <font color="Maroon" title="c8">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">Q</font> or  ex <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E24">A7</font></span></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/>


</div>
