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<span class="kw">let </span><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V8" title="AFF_3:attr.8">satisfying_DES2_3</a>  )</span><br/>

<a NAME="E1:16"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V8" title="AFF_3:attr.8">satisfying_DES2_3</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V8" title="AFF_3:attr.8">satisfying_DES2_3</a> </span><br/>

<span class="kw">thus </span><a NAME="E2:16_1"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V8" title="AFF_3:attr.8">satisfying_DES2_3</a> 
  <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="16_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="aff_3.html#D8" target="_self" title="AFF_3:def.8">AFF_3:def 8</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">P</font>, <font color="Olive" title="b2">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">c9</font>, <font color="Olive" title="b9">p</font>, <font color="Olive" title="b10">q</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">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">P</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b4">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b10">q</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">c9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b9">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b4">a9</font>,<font color="Olive" title="b9">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b7">c</font>,<font color="Olive" title="b10">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b8">c9</font>,<font color="Olive" title="b10">q</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">a9</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b9</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">c9</font>, <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">q</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">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b2">a9</font>,<font color="Olive" title="b6">c9</font> &amp; <font color="Olive" title="b7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">q</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b2">a9</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b8">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b6">c9</font>,<font color="Olive" title="b8">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a9</font>,<font color="Olive" title="b6">c9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">b9</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">c9</font>, <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c11">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> implies <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> )</span><br/>





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:16_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:16_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E4:16_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E6:16_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E7:16_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E8:16_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E9:16_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E10:16_1_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E11:16_1_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E12:16_1_1"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E13:16_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E14:16_1_1"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">and </span><br/><a NAME="E15:16_1_1"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E16:16_1_1"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E17:16_1_1"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E18:16_1_1"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E19:16_1_1"/><span class="lab"><font color="Green" title="E29">A21</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="c6">a9</font>,<font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E20:16_1_1"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E21:16_1_1"/><span class="lab"><font color="Green" title="E31">A23</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="c10">c9</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E22:16_1_1"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E23:16_1_1"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</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">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">A9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>);<br/><span class="kw">set </span><font color="Maroon" title="c14">P9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>);<br/><span class="kw">set </span><font color="Maroon" title="c15">C9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>);<br/><a NAME="E1:16_1_1_1"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E2:16_1_1_1"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/><a NAME="E3:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A28</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E4:16_1_1_1"/><span class="lab"><font color="Green" title="E37">A29</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E3:16_1_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</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="E4:16_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A31</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</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="E5:16_1_1_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E6:16_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E4:16_1_1_1_1"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E7:16_1_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E8:16_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E9:16_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E10:16_1_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E11:16_1_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c6">a9</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="E12:16_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A32</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E13:16_1_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E1:16_1_1_1_1"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E14:16_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E12:16_1_1_1_1"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:16_1_1_1"/><span class="lab"><font color="Green" title="E38">A33</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="E2:16_1_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:16_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A34</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c5">a</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="E4:16_1_1_1_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E4:16_1_1_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E3:16_1_1_1_2"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</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="E6:16_1_1_1"/><span class="lab"><font color="Green" title="E39">A35</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1_1_1_3"/><span class="lab"><font color="Green" title="E40">A36</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:16_1_1_1_3"/><span class="lab"><font color="Green" title="E41">A37</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</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="E3:16_1_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E4:16_1_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c5">a</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="E5:16_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A38</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E6:16_1_1_1_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E2:16_1_1_1_3"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E7:16_1_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E1:16_1_1_1_3"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E8:16_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A39</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E9:16_1_1_1_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E10:16_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A40</font></span>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_1_1_3"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E11:16_1_1_1_3"/>
 not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E5:16_1_1_1_3"><span class="lab"><font color="Green" title="E42">A38</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E12:16_1_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E10:16_1_1_1_3"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E13:16_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</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="E14:16_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E15:16_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</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="E16:16_1_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</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="E17:16_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E5:16_1_1_1_3"><span class="lab"><font color="Green" title="E42">A38</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E18:16_1_1_1_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E19:16_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E5:16_1_1_1_3"><span class="lab"><font color="Green" title="E42">A38</font></span></a>, <a class="txt" href="#E17:16_1_1_1_3"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:16_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E5:16_1_1_1_3"><span class="lab"><font color="Green" title="E42">A38</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:16_1_1_1"/><span class="lab"><font color="Green" title="E40">A42</font></span>: 
<font color="Maroon" title="c7">b</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="16_1_1_1_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1_1_1_4"/><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E2:16_1_1_1_4"/><span class="lab"><font color="Green" title="E42">A44</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E3:16_1_1_1_4"/><span class="lab"><font color="Green" title="E43">A45</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:16_1_1_1_4"/><span class="lab"><font color="Green" title="E44">A46</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:16_1_1_1_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E6:16_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A47</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E3:16_1_1_1_4"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E7:16_1_1_1_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E4:16_1_1_1_4"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E8:16_1_1_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_4"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E9:16_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A48</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E6:16_1_1_1_4"><span class="lab"><font color="Green" title="E45">A47</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E10:16_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E20:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E6:16_1_1_1_4"><span class="lab"><font color="Green" title="E45">A47</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E11:16_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E6:16_1_1_1_4"><span class="lab"><font color="Green" title="E45">A47</font></span></a>, <a class="txt" href="#E9:16_1_1_1_4"><span class="lab"><font color="Green" title="E46">A48</font></span></a>, <a class="txt" href="#E1:16_1_1_1_4"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a>, <a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:16_1_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E9:16_1_1_1_4"><span class="lab"><font color="Green" title="E46">A48</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A49</font></span>: 
<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E9:16_1_1_1"/><span class="lab"><font color="Green" title="E42">A50</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/><a NAME="E10:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A51</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E11:16_1_1_1"/><span class="lab"><font color="Green" title="E44">A52</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_1"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E12:16_1_1_1"/><span class="lab"><font color="Green" title="E45">A53</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E13:16_1_1_1"/><span class="lab"><font color="Green" title="E46">A54</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_5"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">a</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="E2:16_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A55</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</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="E3:16_1_1_1_5"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</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="E4:16_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E12:16_1_1_1"><span class="lab"><font color="Green" title="E45">A53</font></span></a>, <a class="txt" href="#E2:16_1_1_1_5"><span class="lab"><font color="Green" title="E47">A55</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_1_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</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><a NAME="E14:16_1_1_1"/><span class="lab"><font color="Green" title="E47">A56</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E15:16_1_1_1"/><span class="lab"><font color="Green" title="E48">A57</font></span>: 
<font color="Maroon" title="c6">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="c6">a9</font>,<font color="Maroon" title="c10">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_1"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E16:16_1_1_1"/><span class="lab"><font color="Green" title="E49">A58</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E17:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A59</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E9:16_1_1_1"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E11:16_1_1_1"><span class="lab"><font color="Green" title="E44">A52</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E18:16_1_1_1"/><span class="lab"><font color="Green" title="E51">A60</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_6"/><span class="lab"><font color="Green" title="E52">A61</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">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="E2:16_1_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E11:16_1_1_1"><span class="lab"><font color="Green" title="E44">A52</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E3:16_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A62</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E4:16_1_1_1_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E16:16_1_1_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E1:16_1_1_1_6"><span class="lab"><font color="Green" title="E52">A61</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E5:16_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16_1_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E3:16_1_1_1_6"><span class="lab"><font color="Green" title="E53">A62</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E19:16_1_1_1"/><span class="lab"><font color="Green" title="E52">A63</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E20:16_1_1_1"/><span class="lab"><font color="Green" title="E53">A64</font></span>: 
<font color="Maroon" title="c11">p</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="c11">p</font>,<font color="Maroon" title="c12">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E21:16_1_1_1"/><span class="lab"><font color="Green" title="E54">A65</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_7"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</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="E2:16_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A66</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E16:16_1_1_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E20:16_1_1_1"><span class="lab"><font color="Green" title="E53">A64</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E3:16_1_1_1_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:16_1_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E2:16_1_1_1_7"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</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="E22:16_1_1_1"/><span class="lab"><font color="Green" title="E55">A67</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E23:16_1_1_1"/><span class="lab"><font color="Green" title="E56">A68</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_8"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">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="E2:16_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A69</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_1_1_1"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E20:16_1_1_1"><span class="lab"><font color="Green" title="E53">A64</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E11:16_1_1_1"><span class="lab"><font color="Green" title="E44">A52</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E3:16_1_1_1_8"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E4:16_1_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E2:16_1_1_1_8"><span class="lab"><font color="Green" title="E57">A69</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E5:16_1_1_1_8"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E6:16_1_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16_1_1_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</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><a NAME="E24:16_1_1_1"/><span class="lab"><font color="Green" title="E57">A70</font></span>: 
<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><a NAME="E25:16_1_1_1"/><span class="lab"><font color="Green" title="E58">A71</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_9"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E59">A72</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_9_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_1_1_9_1"/>
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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="E2:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c6">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:16_1_1_1"><span class="lab"><font color="Green" title="E57">A70</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/><a NAME="E3:16_1_1_1_9_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c11">p</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="E4:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</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="c6">a9</font>,<font color="Maroon" title="c10">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_1"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E11:16_1_1_1"><span class="lab"><font color="Green" title="E44">A52</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><a NAME="E5:16_1_1_1_9_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> or <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E34">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="E6:16_1_1_1_9_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E11:16_1_1_1"><span class="lab"><font color="Green" title="E44">A52</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/><a NAME="E7:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><span class="kw">hence </span><a NAME="E8:16_1_1_1_9_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E23">A15</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">assume </span><a NAME="E2:16_1_1_1_9"/><span class="lab"><font color="Green" title="E60">A73</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:16_1_1_1_9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c10">c9</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="E4:16_1_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A74</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E14:16_1_1_1"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="txt" href="#E2:16_1_1_1_9"><span class="lab"><font color="Green" title="E60">A73</font></span></a>, <a class="txt" href="#E1:16_1_1_1_9"><span class="lab"><font color="Green" title="E59">A72</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E5:16_1_1_1_9"/><span class="lab"><font color="Green" title="E62">A75</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</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="E6:16_1_1_1_9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_9"><span class="lab"><font color="Green" title="E60">A73</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E7:16_1_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E4:16_1_1_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E19:16_1_1_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a>, <a class="txt" href="#E5:16_1_1_1_9"><span class="lab"><font color="Green" title="E62">A75</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:16_1_1_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E7:16_1_1_1"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="txt" href="#E4:16_1_1_1_9"><span class="lab"><font color="Green" title="E61">A74</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E26:16_1_1_1"/><span class="lab"><font color="Green" title="E59">A76</font></span>: 
<font color="Maroon" title="c12">q</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="c11">p</font>,<font color="Maroon" title="c12">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/><a NAME="E27:16_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E9:16_1_1_1"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="txt" href="#E22:16_1_1_1"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="txt" href="#E20:16_1_1_1"><span class="lab"><font color="Green" title="E53">A64</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E11:16_1_1_1"><span class="lab"><font color="Green" title="E44">A52</font></span></a>, <a class="txt" href="#E24:16_1_1_1"><span class="lab"><font color="Green" title="E57">A70</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E28:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E22:16_1_1_1"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E16:16_1_1_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E20:16_1_1_1"><span class="lab"><font color="Green" title="E53">A64</font></span></a>, <a class="txt" href="#E26:16_1_1_1"><span class="lab"><font color="Green" title="E59">A76</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E11:16_1_1_1"><span class="lab"><font color="Green" title="E44">A52</font></span></a>, <a class="txt" href="#E7:16_1_1_1"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="txt" href="#E21:16_1_1_1"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="txt" href="#E18:16_1_1_1"><span class="lab"><font color="Green" title="E51">A60</font></span></a>, <a class="txt" href="#E23:16_1_1_1"><span class="lab"><font color="Green" title="E56">A68</font></span></a>, <a class="txt" href="#E17:16_1_1_1"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="txt" href="#E8:16_1_1_1"><span class="lab"><font color="Green" title="E41">A49</font></span></a>, <a class="txt" href="#E12:16_1_1_1"><span class="lab"><font color="Green" title="E45">A53</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E19:16_1_1_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a>, <a class="txt" href="#E14:16_1_1_1"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="txt" href="#E13:16_1_1_1"><span class="lab"><font color="Green" title="E46">A54</font></span></a>, <a class="txt" href="#E25:16_1_1_1"><span class="lab"><font color="Green" title="E58">A71</font></span></a>, <a class="ref" href="aff_3.html#D6" target="_self" title="AFF_3:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E29:16_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</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">hence </span><a NAME="E25:16_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 ; <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><span class="kw">end;</span></div>
<a NAME="E2:16"/>
( <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V8" title="AFF_3:attr.8">satisfying_DES2_3</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E10">A77</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V8" title="AFF_3:attr.8">satisfying_DES2_3</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a> </span><br/>

<span class="kw">thus </span><a NAME="E2:16_2"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a> 
  <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="16_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="aff_3.html#D6" target="_self" title="AFF_3:def.6">AFF_3:def 6</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">P</font>, <font color="Olive" title="b2">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">c9</font>, <font color="Olive" title="b9">p</font>, <font color="Olive" title="b10">q</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">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">C</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b4">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b10">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b9">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b4">a9</font>,<font color="Olive" title="b9">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b7">c</font>,<font color="Olive" title="b10">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b8">c9</font>,<font color="Olive" title="b10">q</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font> holds <br/><font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b8">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">a9</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b9</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">c9</font>, <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">q</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">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b2">a9</font>,<font color="Olive" title="b6">c9</font> &amp; <font color="Olive" title="b7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b2">a9</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b8">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b6">c9</font>,<font color="Olive" title="b8">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a9</font>,<font color="Olive" title="b6">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">b9</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">c9</font>, <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c11">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> )</span><br/>





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_2_1"/><span class="lab"><font color="Green" title="E11">A78</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:16_2_1"/><span class="lab"><font color="Green" title="E12">A79</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:16_2_1"/><span class="lab"><font color="Green" title="E13">A80</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E4:16_2_1"/><span class="lab"><font color="Green" title="E14">A81</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E5:16_2_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E6:16_2_1"/><span class="lab"><font color="Green" title="E15">A82</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E7:16_2_1"/><span class="lab"><font color="Green" title="E16">A83</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E8:16_2_1"/><span class="lab"><font color="Green" title="E17">A84</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E9:16_2_1"/><span class="lab"><font color="Green" title="E18">A85</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E10:16_2_1"/><span class="lab"><font color="Green" title="E19">A86</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E11:16_2_1"/><span class="lab"><font color="Green" title="E20">A87</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E12:16_2_1"/><span class="lab"><font color="Green" title="E21">A88</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E13:16_2_1"/><span class="lab"><font color="Green" title="E22">A89</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E14:16_2_1"/><span class="lab"><font color="Green" title="E23">A90</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E15:16_2_1"/><span class="lab"><font color="Green" title="E24">A91</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">and </span><br/><a NAME="E16:16_2_1"/><span class="lab"><font color="Green" title="E25">A92</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E17:16_2_1"/><span class="lab"><font color="Green" title="E26">A93</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E18:16_2_1"/><span class="lab"><font color="Green" title="E27">A94</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E19:16_2_1"/><span class="lab"><font color="Green" title="E28">A95</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="c6">a9</font>,<font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E20:16_2_1"/><span class="lab"><font color="Green" title="E29">A96</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E21:16_2_1"/><span class="lab"><font color="Green" title="E30">A97</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="c10">c9</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E22:16_2_1"/><span class="lab"><font color="Green" title="E31">A98</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font></span><br/>

<a NAME="E23:16_2_1"/><span class="lab"><font color="Green" title="E32">A99</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E22">A89</font></span></a>, <a class="txt" href="#E14:16_2_1"><span class="lab"><font color="Green" title="E23">A90</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E24:16_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A100</font></span>: 
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E10:16_2_1"><span class="lab"><font color="Green" title="E19">A86</font></span></a>, <a class="txt" href="#E11:16_2_1"><span class="lab"><font color="Green" title="E20">A87</font></span></a>, <a class="txt" href="#E12:16_2_1"><span class="lab"><font color="Green" title="E21">A88</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">assume </span><a NAME="E25:16_2_1"/><span class="lab"><font color="Green" title="E34">A101</font></span>: 
 not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">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="E26:16_2_1"/><span class="lab"><font color="Green" title="E35">A102</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2_1_1"/><span class="lab"><font color="Green" title="E36">A103</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_2_1"><span class="lab"><font color="Green" title="E31">A98</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E3:16_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</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="E4:16_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A104</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</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="E5:16_2_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E6:16_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E24">A91</font></span></a>, <a class="txt" href="#E4:16_2_1_1"><span class="lab"><font color="Green" title="E37">A104</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E7:16_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_2_1"><span class="lab"><font color="Green" title="E28">A95</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E8:16_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A105</font></span>: 
( <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2_1"><span class="lab"><font color="Green" title="E11">A78</font></span></a>, <a class="txt" href="#E7:16_2_1"><span class="lab"><font color="Green" title="E16">A83</font></span></a>, <a class="txt" href="#E8:16_2_1"><span class="lab"><font color="Green" title="E17">A84</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E9:16_2_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_2_1"><span class="lab"><font color="Green" title="E30">A97</font></span></a>, <a class="txt" href="#E1:16_2_1_1"><span class="lab"><font color="Green" title="E36">A103</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E10:16_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font> or <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2_1"><span class="lab"><font color="Green" title="E13">A80</font></span></a>, <a class="txt" href="#E11:16_2_1"><span class="lab"><font color="Green" title="E20">A87</font></span></a>, <a class="txt" href="#E12:16_2_1"><span class="lab"><font color="Green" title="E21">A88</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:16_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A81</font></span></a>, <a class="txt" href="#E6:16_2_1"><span class="lab"><font color="Green" title="E15">A82</font></span></a>, <a class="txt" href="#E10:16_2_1"><span class="lab"><font color="Green" title="E19">A86</font></span></a>, <a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E22">A89</font></span></a>, <a class="txt" href="#E25:16_2_1"><span class="lab"><font color="Green" title="E34">A101</font></span></a>, <a class="txt" href="#E23:16_2_1"><span class="lab"><font color="Green" title="E32">A99</font></span></a>, <a class="txt" href="#E8:16_2_1_1"><span class="lab"><font color="Green" title="E38">A105</font></span></a>, <a class="ref" href="aff_1.html#T2" title="AFF_1:th.2">AFF_1:2</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>
<a NAME="E27:16_2_1"/><span class="lab"><font color="Green" title="E36">A106</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="E2:16_2_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_2_1"><span class="lab"><font color="Green" title="E31">A98</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:16_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A107</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c5">a</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="E4:16_2_1_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_2_1"><span class="lab"><font color="Green" title="E29">A96</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E24">A91</font></span></a>, <a class="txt" href="#E26:16_2_1"><span class="lab"><font color="Green" title="E35">A102</font></span></a>, <a class="txt" href="#E3:16_2_1_2"><span class="lab"><font color="Green" title="E37">A107</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</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="E28:16_2_1"/><span class="lab"><font color="Green" title="E37">A108</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_2_1_3"/><span class="lab"><font color="Green" title="E38">A109</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:16_2_1_3"/><span class="lab"><font color="Green" title="E39">A110</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</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="E3:16_2_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_2_1"><span class="lab"><font color="Green" title="E28">A95</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E4:16_2_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_2_1"><span class="lab"><font color="Green" title="E12">A79</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E10:16_2_1"><span class="lab"><font color="Green" title="E19">A86</font></span></a>, <a class="txt" href="#E27:16_2_1"><span class="lab"><font color="Green" title="E36">A106</font></span></a>, <a class="txt" href="#E1:16_2_1_3"><span class="lab"><font color="Green" title="E38">A109</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E5:16_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A111</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A81</font></span></a>, <a class="txt" href="#E7:16_2_1"><span class="lab"><font color="Green" title="E16">A83</font></span></a>, <a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E22">A89</font></span></a>, <a class="txt" href="#E21:16_2_1"><span class="lab"><font color="Green" title="E30">A97</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E6:16_2_1_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_2_1"><span class="lab"><font color="Green" title="E29">A96</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E7:16_2_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font> or <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2_1"><span class="lab"><font color="Green" title="E13">A80</font></span></a>, <a class="txt" href="#E11:16_2_1"><span class="lab"><font color="Green" title="E20">A87</font></span></a>, <a class="txt" href="#E12:16_2_1"><span class="lab"><font color="Green" title="E21">A88</font></span></a>, <a class="txt" href="#E5:16_2_1_3"><span class="lab"><font color="Green" title="E40">A111</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E8:16_2_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_2_1"><span class="lab"><font color="Green" title="E15">A82</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="txt" href="#E25:16_2_1"><span class="lab"><font color="Green" title="E34">A101</font></span></a>, <a class="txt" href="#E23:16_2_1"><span class="lab"><font color="Green" title="E32">A99</font></span></a>, <a class="txt" href="#E2:16_2_1_3"><span class="lab"><font color="Green" title="E39">A110</font></span></a>, <a class="ref" href="aff_1.html#T2" title="AFF_1:th.2">AFF_1:2</a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E9:16_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E10:16_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16_2_1"><span class="lab"><font color="Green" title="E26">A93</font></span></a>, <a class="txt" href="#E22:16_2_1"><span class="lab"><font color="Green" title="E31">A98</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E11:16_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</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="E12:16_2_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</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="E13:16_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A112</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</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="E14:16_2_1_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E24">A91</font></span></a>, <a class="txt" href="#E27:16_2_1"><span class="lab"><font color="Green" title="E36">A106</font></span></a>, <a class="txt" href="#E13:16_2_1_3"><span class="lab"><font color="Green" title="E41">A112</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</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="E29:16_2_1"/><span class="lab"><font color="Green" title="E38">A113</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2_1_4"/><span class="lab"><font color="Green" title="E39">A114</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</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="E2:16_2_1_4"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E3:16_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2_1"><span class="lab"><font color="Green" title="E11">A78</font></span></a>, <a class="txt" href="#E7:16_2_1"><span class="lab"><font color="Green" title="E16">A83</font></span></a>, <a class="txt" href="#E8:16_2_1"><span class="lab"><font color="Green" title="E17">A84</font></span></a>, <a class="txt" href="#E27:16_2_1"><span class="lab"><font color="Green" title="E36">A106</font></span></a>, <a class="txt" href="#E28:16_2_1"><span class="lab"><font color="Green" title="E37">A108</font></span></a>, <a class="txt" href="#E1:16_2_1_4"><span class="lab"><font color="Green" title="E39">A114</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:16_2_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A81</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E22">A89</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">set </span><font color="Maroon" title="c13">A9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c14">P9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c15">C9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>);<br/>
<a NAME="E30:16_2_1"/><span class="lab"><font color="Green" title="E39">A115</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E24">A91</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E31:16_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A116</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E32:16_2_1"/><span class="lab"><font color="Green" title="E41">A117</font></span>: 
( <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_2_1"><span class="lab"><font color="Green" title="E16">A83</font></span></a>, <a class="txt" href="#E8:16_2_1"><span class="lab"><font color="Green" title="E17">A84</font></span></a>, <a class="txt" href="#E11:16_2_1"><span class="lab"><font color="Green" title="E20">A87</font></span></a>, <a class="txt" href="#E12:16_2_1"><span class="lab"><font color="Green" title="E21">A88</font></span></a>, <a class="txt" href="#E14:16_2_1"><span class="lab"><font color="Green" title="E23">A90</font></span></a>, <a class="txt" href="#E20:16_2_1"><span class="lab"><font color="Green" title="E29">A96</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E33:16_2_1"/><span class="lab"><font color="Green" title="E42">A118</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_2_1"><span class="lab"><font color="Green" title="E28">A95</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E34:16_2_1"/><span class="lab"><font color="Green" title="E43">A119</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:16_2_1"><span class="lab"><font color="Green" title="E39">A115</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E35:16_2_1"/><span class="lab"><font color="Green" title="E44">A120</font></span>: 
<font color="Maroon" title="c7">b</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="16_2_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2_1_5"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A121</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_2_1"><span class="lab"><font color="Green" title="E28">A95</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E3:16_2_1_5"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E4:16_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A122</font></span>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2_1"><span class="lab"><font color="Green" title="E11">A78</font></span></a>, <a class="txt" href="#E7:16_2_1"><span class="lab"><font color="Green" title="E16">A83</font></span></a>, <a class="txt" href="#E8:16_2_1"><span class="lab"><font color="Green" title="E17">A84</font></span></a>, <a class="txt" href="#E28:16_2_1"><span class="lab"><font color="Green" title="E37">A108</font></span></a>, <a class="txt" href="#E2:16_2_1_5"><span class="lab"><font color="Green" title="E45">A121</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E5:16_2_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A81</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E22">A89</font></span></a>, <a class="txt" href="#E20:16_2_1"><span class="lab"><font color="Green" title="E29">A96</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E6:16_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E7:16_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16_2_1"><span class="lab"><font color="Green" title="E26">A93</font></span></a>, <a class="txt" href="#E22:16_2_1"><span class="lab"><font color="Green" title="E31">A98</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E8:16_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</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="E9:16_2_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</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/>
<span class="kw">hence </span><a NAME="E10:16_2_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A81</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E22">A89</font></span></a>, <a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E24">A91</font></span></a>, <a class="txt" href="#E4:16_2_1_5"><span class="lab"><font color="Green" title="E46">A122</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</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>
<a NAME="E36:16_2_1"/><span class="lab"><font color="Green" title="E45">A123</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2_1_6"/><span class="lab"><font color="Green" title="E46">A124</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">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="E2:16_2_1_6"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_2_1"><span class="lab"><font color="Green" title="E29">A96</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E3:16_2_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2_1"><span class="lab"><font color="Green" title="E13">A80</font></span></a>, <a class="txt" href="#E11:16_2_1"><span class="lab"><font color="Green" title="E20">A87</font></span></a>, <a class="txt" href="#E12:16_2_1"><span class="lab"><font color="Green" title="E21">A88</font></span></a>, <a class="txt" href="#E26:16_2_1"><span class="lab"><font color="Green" title="E35">A102</font></span></a>, <a class="txt" href="#E1:16_2_1_6"><span class="lab"><font color="Green" title="E46">A124</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E4:16_2_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A125</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_2_1"><span class="lab"><font color="Green" title="E15">A82</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E21:16_2_1"><span class="lab"><font color="Green" title="E30">A97</font></span></a>, <a class="txt" href="#E23:16_2_1"><span class="lab"><font color="Green" title="E32">A99</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E5:16_2_1_6"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_2_1"><span class="lab"><font color="Green" title="E29">A96</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E6:16_2_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_2_1"><span class="lab"><font color="Green" title="E12">A79</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E10:16_2_1"><span class="lab"><font color="Green" title="E19">A86</font></span></a>, <a class="txt" href="#E26:16_2_1"><span class="lab"><font color="Green" title="E35">A102</font></span></a>, <a class="txt" href="#E35:16_2_1"><span class="lab"><font color="Green" title="E44">A120</font></span></a>, <a class="txt" href="#E4:16_2_1_6"><span class="lab"><font color="Green" title="E47">A125</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16_2_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_2_1"><span class="lab"><font color="Green" title="E15">A82</font></span></a>, <a class="txt" href="#E11:16_2_1"><span class="lab"><font color="Green" title="E20">A87</font></span></a>, <a class="txt" href="#E23:16_2_1"><span class="lab"><font color="Green" title="E32">A99</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>
<a NAME="E37:16_2_1"/><span class="lab"><font color="Green" title="E46">A126</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">b9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="txt" href="#E21:16_2_1"><span class="lab"><font color="Green" title="E30">A97</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E38:16_2_1"/><span class="lab"><font color="Green" title="E47">A127</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_2_1"><span class="lab"><font color="Green" title="E25">A92</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E39:16_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A128</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E40:16_2_1"/><span class="lab"><font color="Green" title="E49">A129</font></span>: 
<font color="Maroon" title="c11">p</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="c11">p</font>,<font color="Maroon" title="c12">q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16_2_1"><span class="lab"><font color="Green" title="E26">A93</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E41:16_2_1"/><span class="lab"><font color="Green" title="E50">A130</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2_1_7"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</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="E2:16_2_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A131</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:16_2_1"><span class="lab"><font color="Green" title="E40">A116</font></span></a>, <a class="txt" href="#E34:16_2_1"><span class="lab"><font color="Green" title="E43">A119</font></span></a>, <a class="txt" href="#E40:16_2_1"><span class="lab"><font color="Green" title="E49">A129</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E3:16_2_1_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E27">A94</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:16_2_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E24">A91</font></span></a>, <a class="txt" href="#E27:16_2_1"><span class="lab"><font color="Green" title="E36">A106</font></span></a>, <a class="txt" href="#E2:16_2_1_7"><span class="lab"><font color="Green" title="E51">A131</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</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="E42:16_2_1"/><span class="lab"><font color="Green" title="E51">A132</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c12">q</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="c11">p</font>,<font color="Maroon" title="c12">q</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16_2_1"><span class="lab"><font color="Green" title="E26">A93</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E43:16_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A133</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16_2_1"><span class="lab"><font color="Green" title="E26">A93</font></span></a>, <a class="txt" href="#E22:16_2_1"><span class="lab"><font color="Green" title="E31">A98</font></span></a>, <a class="txt" href="#E30:16_2_1"><span class="lab"><font color="Green" title="E39">A115</font></span></a>, <a class="txt" href="#E31:16_2_1"><span class="lab"><font color="Green" title="E40">A116</font></span></a>, <a class="txt" href="#E34:16_2_1"><span class="lab"><font color="Green" title="E43">A119</font></span></a>, <a class="txt" href="#E40:16_2_1"><span class="lab"><font color="Green" title="E49">A129</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E44:16_2_1"/><span class="lab"><font color="Green" title="E53">A134</font></span>: 
( <font color="Maroon" title="c6">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="c6">a9</font>,<font color="Maroon" title="c10">c9</font>) &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:16_2_1"><span class="lab"><font color="Green" title="E47">A127</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E45:16_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16_2_1"><span class="lab"><font color="Green" title="E34">A101</font></span></a>, <a class="txt" href="#E31:16_2_1"><span class="lab"><font color="Green" title="E40">A116</font></span></a>, <a class="txt" href="#E34:16_2_1"><span class="lab"><font color="Green" title="E43">A119</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>;<br/>
<a NAME="E46:16_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E10">A77</font></span></a>, <a class="txt" href="#E38:16_2_1"><span class="lab"><font color="Green" title="E47">A127</font></span></a>, <a class="txt" href="#E31:16_2_1"><span class="lab"><font color="Green" title="E40">A116</font></span></a>, <a class="txt" href="#E39:16_2_1"><span class="lab"><font color="Green" title="E48">A128</font></span></a>, <a class="txt" href="#E34:16_2_1"><span class="lab"><font color="Green" title="E43">A119</font></span></a>, <a class="txt" href="#E40:16_2_1"><span class="lab"><font color="Green" title="E49">A129</font></span></a>, <a class="txt" href="#E42:16_2_1"><span class="lab"><font color="Green" title="E51">A132</font></span></a>, <a class="txt" href="#E44:16_2_1"><span class="lab"><font color="Green" title="E53">A134</font></span></a>, <a class="txt" href="#E24:16_2_1"><span class="lab"><font color="Green" title="E33">A100</font></span></a>, <a class="txt" href="#E43:16_2_1"><span class="lab"><font color="Green" title="E52">A133</font></span></a>, <a class="txt" href="#E41:16_2_1"><span class="lab"><font color="Green" title="E50">A130</font></span></a>, <a class="txt" href="#E35:16_2_1"><span class="lab"><font color="Green" title="E44">A120</font></span></a>, <a class="txt" href="#E36:16_2_1"><span class="lab"><font color="Green" title="E45">A123</font></span></a>, <a class="txt" href="#E29:16_2_1"><span class="lab"><font color="Green" title="E38">A113</font></span></a>, <a class="txt" href="#E32:16_2_1"><span class="lab"><font color="Green" title="E41">A117</font></span></a>, <a class="txt" href="#E37:16_2_1"><span class="lab"><font color="Green" title="E46">A126</font></span></a>, <a class="txt" href="#E33:16_2_1"><span class="lab"><font color="Green" title="E42">A118</font></span></a>, <a class="ref" href="aff_3.html#D8" target="_self" title="AFF_3:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:16_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16_2_1"><span class="lab"><font color="Green" title="E34">A101</font></span></a>, <a class="txt" href="#E34:16_2_1"><span class="lab"><font color="Green" title="E43">A119</font></span></a>, <a class="txt" href="#E44:16_2_1"><span class="lab"><font color="Green" title="E53">A134</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:16"/>
( <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V8" title="AFF_3:attr.8">satisfying_DES2_3</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E9">A1</font></span></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>
