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<span class="kw">let </span><font color="Maroon" title="c1">AFP</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">AFP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>  implies  for <font color="Olive" title="b1">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a9</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">c9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp;  not <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b6">a9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> holds <br/> <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b7">b9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">AFP</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a9</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">c9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp;  not <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b6">a9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> holds <br/> <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b7">b9</font></span><br/>

<span class="kw">thus </span><a NAME="E2:3"/>
 for <font color="Olive" title="b1">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a9</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">c9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp;  not <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b6">a9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> holds <br/> <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b7">b9</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_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">AFP</font>; <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">AFP</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="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">AFP</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="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="E1:3_1"/><span class="lab"><font color="Green" title="E5">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="E2:3_1"/><span class="lab"><font color="Green" title="E6">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="E3:3_1"/><span class="lab"><font color="Green" title="E7">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="E4:3_1"/><span class="lab"><font color="Green" title="E8">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="E5:3_1"/><span class="lab"><font color="Green" title="E9">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="E6:3_1"/><span class="lab"><font color="Green" title="E10">A7</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="E7:3_1"/><span class="lab"><font color="Green" title="E11">A8</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="E8:3_1"/><span class="lab"><font color="Green" title="E12">A9</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="E9:3_1"/><span class="lab"><font color="Green" title="E13">A10</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="E10:3_1"/><span class="lab"><font color="Green" title="E14">A11</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="E11:3_1"/><span class="lab"><font color="Green" title="E15">A12</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">  <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">AFP</font><span class="kw"> such that </span><br/><a NAME="E13:3_1"/><span class="lab"><font color="Green" title="E16">A13</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="E14:3_1"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/>
<a NAME="E15:3_1"/><span class="lab"><font color="Green" title="E18">A15</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="#E14:3_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</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="E16:3_1"/><span class="lab"><font color="Green" title="E19">A16</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="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E17:3_1"/><span class="lab"><font color="Green" title="E20">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="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#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E18:3_1"/><span class="lab"><font color="Green" title="E21">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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E19:3_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#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E20:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">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="E21:3_1"/><span class="lab"><font color="Green" title="E23">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="#E5:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="aff_1.html#T35" title="AFF_1:th.35">AFF_1:35</a></span>;<br/>
<a NAME="E22:3_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#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c10">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:3_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#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c10">P</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="E2:3_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#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:3_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E21:3_1"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="txt" href="#E16:3_1"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E17:3_1"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="c13">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E24:3_1"/><span class="lab"><font color="Green" title="E24">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="E25:3_1"/><span class="lab"><font color="Green" title="E25">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="#E20:3_1"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="txt" href="#E15:3_1"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E26:3_1"/><span class="lab"><font color="Green" title="E26">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>)
 
<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="E27:3_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="#E11:3_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E14:3_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E28:3_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="#E6:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E8:3_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E29:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A24</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="#E13:3_1"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E15:3_1"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<a NAME="E30:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A25</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="#E15:3_1"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E25:3_1"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E31:3_1"/><span class="lab"><font color="Green" title="E29">A26</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="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E32:3_1"/><span class="lab"><font color="Green" title="E30">A27</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="#E20:3_1"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="txt" href="#E16:3_1"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E17:3_1"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E24:3_1"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E33:3_1"/><span class="lab"><font color="Green" title="E31">A28</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="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">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="E34:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A29</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="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E7:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E26:3_1"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E31:3_1"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E35:3_1"/><span class="lab"><font color="Green" title="E33">A30</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="3_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:3_1_2"/><span class="lab"><font color="Green" title="E34">A31</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="#E10:3_1"><span class="lab"><font color="Green" title="E14">A11</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="E2:3_1_2"/><span class="lab"><font color="Green" title="E35">A32</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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:3_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="#E4:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E14:3_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E29:3_1"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E4:3_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="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E33:3_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="txt" href="#E26:3_1"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E31:3_1"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="txt" href="#E34:3_1"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="txt" href="#E13:3_1"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E5:3_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="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E2:3_1_2"><span class="lab"><font color="Green" title="E35">A32</font></span></a>, <a class="txt" href="#E1:3_1_2"><span class="lab"><font color="Green" title="E34">A31</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="E6:3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_1"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</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="E36:3_1"/><span class="lab"><font color="Green" title="E34">A33</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="3_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:3_1_3"/><span class="lab"><font color="Green" title="E35">A34</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/>

<div><span class="lab"><font color="Green" title="E36">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:3_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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:3_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="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E33:3_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="txt" href="#E26:3_1"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E31:3_1"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="txt" href="#E34:3_1"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="txt" href="#E1:3_1_3"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:3_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_1"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</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="E3:3_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="#E9:3_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E10:3_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E1:3_1_3"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E4:3_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="#E2:3_1_3"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E5:3_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="E6:3_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#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E7:3_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="E8:3_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#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E9:3_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="#E6:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E10:3_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#T34" title="AFF_1:th.34">AFF_1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:3_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">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="E37:3_1"/><span class="lab"><font color="Green" title="E35">A36</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="#E3:3_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="aff_1.html#T35" title="AFF_1:th.35">AFF_1:35</a></span>;<br/>
<a NAME="E38:3_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="#E13:3_1"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E14:3_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E25:3_1"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a></span>;<br/>
<a NAME="E39:3_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="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E7:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E10:3_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E32:3_1"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="txt" href="homothet.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<a NAME="E40:3_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="#E9:3_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E37:3_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E41:3_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#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E42:3_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="E43:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A37</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#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E44:3_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="#E13:3_1"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E15:3_1"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E25:3_1"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="txt" href="#E29:3_1"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E45:3_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="#E18:3_1"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="txt" href="#E32:3_1"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="txt" href="#E43:3_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E30:3_1"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E46:3_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="E47:3_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#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E48:3_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="#E9:3_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E35:3_1"><span class="lab"><font color="Green" title="E33">A30</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E49:3_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="#E8:3_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E36:3_1"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E50:3_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="#E6:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E51:3_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#T34" title="AFF_1:th.34">AFF_1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E52:3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E21:3_1"><span class="lab"><font color="Green" title="E23">A20</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>


</div>
