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<span class="kw">let </span><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V10" title="AFF_2:attr.10">satisfying_TDES_3</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V10" title="AFF_2:attr.10">satisfying_TDES_3</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="aff_2.html#D7" target="_self" title="AFF_2:def.7">AFF_2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp;  not <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b7">c9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b7">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a9</font>, <font color="Maroon" title="c8">b9</font>, <font color="Maroon" title="c9">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp;  not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b9</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">a9</font>,<font color="Maroon" title="c8">b9</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font> &amp; <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="c2">K</font> implies <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c2">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:20"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E4:20"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E5:20"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
 not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E7:20"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E8:20"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E9:20"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">and </span><br/><a NAME="E10:20"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">and </span><br/><a NAME="E11:20"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<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">a9</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">and </span><br/><a NAME="E12:20"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font>
 <span class="kw">and </span><br/><a NAME="E13:20"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<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="c2">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">T</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font>);<br/>
<a NAME="E14:20"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c3">o</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">o</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E15:20"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
 not <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E16:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>;<br/>
<a NAME="E17:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<font color="Maroon" title="c5">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="c5">b</font>,<font color="Maroon" title="c6">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E18:20"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E19:20"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c6">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="c5">b</font>,<font color="Maroon" title="c6">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E20:20"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c4">a</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">o</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E21:20"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</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="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E22:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c7">a9</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">o</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E23:20"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="aff_1.html#T35" title="AFF_1:th.35">AFF_1:35</a></span>;<br/>
<a NAME="E24:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A24</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</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#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E25:20"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c5">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">o</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E26:20"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">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="c3">o</font>,<font color="Maroon" title="c5">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_1"/><span class="kw">then </span>
<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>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a>, <a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:20"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
<font color="Maroon" title="c3">o</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">o</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E28:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c8">b9</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">o</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E29:20"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:20_2"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:20_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E4:20_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:20"/><span class="lab"><font color="Green" title="E44">A32</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_3"/>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A33</font></span>: 
<font color="Maroon" title="c7">a9</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">o</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E3:20_3"/>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E4:20_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</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">o</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E2:20_3"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E31:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A34</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c9">c9</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="c7">a9</font>,<font color="Maroon" title="c9">c9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E32:20"/><span class="lab"><font color="Green" title="E46">A35</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">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="txt" href="#E16:20"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E34:20"/><span class="lab"><font color="Green" title="E47">A36</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">N</font>
 <span class="kw">and </span><br/><a NAME="E35:20"/><span class="lab"><font color="Green" title="E48">A37</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c14">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/>
<a NAME="E36:20"/><span class="lab"><font color="Green" title="E49">A38</font></span>: 
<font color="Maroon" title="c14">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="#E35:20"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<a NAME="E37:20"/><span class="lab"><font color="Green" title="E50">A39</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_4"/><span class="kw">then </span>
<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="c4">a</font>,<font color="Maroon" title="c6">c</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="E3:20_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E4:20_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T34" title="AFF_1:th.34">AFF_1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E38:20"/>
 not <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_5"/><span class="lab"><font color="Green" title="E51">A40</font></span>: 
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_5"/>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E3:20_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E1:20_5"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E4:20_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E5:20_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T34" title="AFF_1:th.34">AFF_1:34</a></span>;<br/>
<a NAME="E6:20_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</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="c5">b</font>,<font color="Maroon" title="c6">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E46">A35</font></span></a>, <a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:20"><span class="lab"><font color="Green" title="E50">A39</font></span></a>, <a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E46">A35</font></span></a>, <a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E40:20"/><span class="lab"><font color="Green" title="E51">A41</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">N</font>
 <span class="kw">and </span><br/><a NAME="E41:20"/><span class="lab"><font color="Green" title="E52">A42</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E42:20"/><span class="lab"><font color="Green" title="E53">A43</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="txt" href="#E40:20"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E43:20"/>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:20"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E44:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E45:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A44</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="aff_2.html#D10" target="_self" title="AFF_2:def.10">Def10</a></span>;<br/>
<a NAME="E46:20"/><span class="lab"><font color="Green" title="E55">A45</font></span>: 
<font color="Maroon" title="c7">a9</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="c7">a9</font>,<font color="Maroon" title="c9">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E47:20"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</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="c7">a9</font>,<font color="Maroon" title="c9">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E31:20"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E41:20"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E48:20"/><span class="kw">then </span>
<font color="Maroon" title="c2">K</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">a9</font>,<font color="Maroon" title="c9">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E31:20"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E49:20"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</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">o</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E46:20"><span class="lab"><font color="Green" title="E55">A45</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E50:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</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">o</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E51:20"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
