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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="Red">::  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>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V8" title="AFF_2:attr.8">satisfying_TDES_1</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</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="c1">AP</font> is  <a href="aff_2.html#V8" title="AFF_2:attr.8">satisfying_TDES_1</a> </span><br/>

<span class="kw">thus </span><a NAME="E2:31"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V8" title="AFF_2:attr.8">satisfying_TDES_1</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<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="Red">:: according to </font></span><a class="ref" href="aff_2.html#D8" target="_self" title="AFF_2:def.8">AFF_2:def 8</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">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; <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="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> &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/> <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></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="Red">::  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; <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="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> &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  <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><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_1"/><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="E2:31_1"/><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="E3:31_1"/><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="E4:31_1"/><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="E5:31_1"/><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="E6:31_1"/><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="E7:31_1"/><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="E8:31_1"/><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="E9:31_1"/><span class="lab"><font color="Green" title="E24">A10</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="E10:31_1"/><span class="lab"><font color="Green" title="E25">A11</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="c9">c9</font>
 <span class="kw">and </span><br/><a NAME="E11:31_1"/><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="E12:31_1"/><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="Red">::  thesis: </font></span></a><span class="hide">  <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><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">P</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="E14:31_1"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">and </span><br/><a NAME="E15:31_1"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E16:31_1"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c10">P</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="#E15:31_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">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="c5">b</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">C</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/>
<a NAME="E17:31_1"/><span class="lab"><font color="Green" title="E31">A17</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> &amp; <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="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E18:31_1"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
not  <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E19:31_1"/><span class="kw">then </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="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E20:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A19</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#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E21:31_1"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
not <font color="Maroon" title="c5">b</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="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E12:31_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/>
<a NAME="E22:31_1"/>
not  <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 href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1_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="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1_1"/><span class="kw">then </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> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E21:31_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E17:31_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</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">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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="E24:31_1"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c13">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="c3">o</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E25:31_1"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:31_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E16:31_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<a NAME="E26:31_1"/><span class="lab"><font color="Green" title="E37">A23</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> &amp; <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="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E27:31_1"/><span class="lab"><font color="Green" title="E38">A24</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="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:31_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E17:31_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E24:31_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E28:31_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:31_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E15:31_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></span>;<br/>
<a NAME="E29:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E9:31_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/>
<a NAME="E30:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:31_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E16:31_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<a NAME="E31:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A26</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E25:31_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E32:31_1"/><span class="lab"><font color="Green" title="E41">A27</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="#E2:31_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E33:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A28</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="#E2:31_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E8:31_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E26:31_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E34:31_1"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c8">b9</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="31_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_1_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="#E11:31_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:31_1_2"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c8">b9</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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:31_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E15:31_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E30:31_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E4:31_1_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="c3">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E32:31_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E26:31_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E33:31_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E14:31_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E5:31_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E2:31_1_2"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E1:31_1_2"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</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>
<a NAME="E35:31_1"/><span class="lab"><font color="Green" title="E44">A32</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="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E12:31_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/>
<a NAME="E36:31_1"/>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c13">x</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="#E14:31_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E15:31_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E25:31_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></span>;<br/>
<a NAME="E37:31_1"/><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="c7">a9</font>,<font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E12:31_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E38:31_1"/><span class="kw">then </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="c13">x</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E6:31_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E8:31_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E11:31_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E12:31_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E27:31_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="aff_2.html#D7" target="_self" title="AFF_2:def.7">Def7</a></span>;<br/>
<a NAME="E39:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E35:31_1"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E40:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E41:31_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c13">x</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="E42:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A33</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E43:31_1"/><span class="lab"><font color="Green" title="E46">A34</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="31_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1_3"/><span class="lab"><font color="Green" title="E47">A35</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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><span class="lab"><font color="Green" title="E48">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_1_3_1"/>
<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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:31_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E32:31_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E26:31_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E33:31_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E1:31_1_3"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</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>
<a NAME="E3:31_1_3"/>
( <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="c5">b</font>,<font color="Maroon" title="c6">c</font> or <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="#E10:31_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E11:31_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E1:31_1_3"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:31_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_3"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:31_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</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="E6:31_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E7:31_1_3"/><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="c4">a</font>,<font color="Maroon" title="c5">b</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="E8:31_1_3"/><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#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E9:31_1_3"/><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="#E7:31_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:31_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/>
<a NAME="E10:31_1_3"/><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#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:31_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</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>
<a NAME="E44:31_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:31_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E16:31_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E25:31_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E30:31_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E45:31_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E27:31_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E42:31_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E31:31_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<a NAME="E46:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c7">a9</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="E47:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</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="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E48:31_1"/><span class="kw">then </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="c7">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E34:31_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E49:31_1"/><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="c5">b</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E43:31_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E50:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</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="#E7:31_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:31_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/>
<a NAME="E51:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</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>
 
<span class="kw">by</span> <span class="lab"><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="E52:31_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E21:31_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</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>


</div>
