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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><i><font color="Green" title="E9">A1</font></i>: 
( <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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E10">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">b'</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">c'</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">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="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">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="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">c'</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#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Olive" title="b4">a'</font>,<font color="Olive" title="b8">c'</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">c'</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">b'</font>,<font color="Olive" title="b4">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="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">b'</font>,<font color="Olive" title="b8">c'</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">a'</font>,<font color="Olive" title="b8">c'</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">C</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">a'</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b'</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">c'</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">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="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">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="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">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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Olive" title="b2">a'</font>,<font color="Olive" title="b6">c'</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">c'</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">b'</font>,<font color="Olive" title="b2">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="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">b'</font>,<font color="Olive" title="b6">c'</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">a'</font>,<font color="Olive" title="b6">c'</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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font></span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">b'</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">c'</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:()"><i><font color="Red">::  thesis: </font></i></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">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="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">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="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">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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</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">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="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">b'</font>,<font color="Maroon" title="c6">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="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">b'</font>,<font color="Maroon" title="c10">c'</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">a'</font>,<font color="Maroon" title="c10">c'</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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font> )</span><br/>





















<b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
( <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">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="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">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="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">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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</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">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="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">b'</font>,<font color="Maroon" title="c6">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="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">b'</font>,<font color="Maroon" title="c10">c'</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">a'</font>,<font color="Maroon" title="c10">c'</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:16_1_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <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> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c8">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> )
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><b>set </b><font color="Maroon" title="c13">A'</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/><b>set </b><font color="Maroon" title="c14">P'</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/><b>set </b><font color="Maroon" title="c15">C'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font>;<br/><a NAME="E2:16_1_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
(  <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;  <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;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">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> &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> &amp; <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> &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> &amp; <font color="Maroon" title="c6">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="c6">a'</font>,<font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> )
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:16_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/><a NAME="E3:16_1_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
<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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:16_1_1_1_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:16_1_1_1_1"/><b>then </b>
(  <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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E5:16_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E6:16_1_1_1_1"/><b>then </b>
 <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">a'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E7:16_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">b'</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">a'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E8:16_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E9:16_1_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E10:16_1_1_1_1"/><b>then </b>
 <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">c'</font>,<font color="Maroon" title="c6">a'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E11:16_1_1_1_1"/><b>then </b><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E12:16_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">c'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:16_1_1_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E13:16_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E14:16_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E11:16_1_1_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:16_1_1_1"/><i><font color="Green" title="E15">A9</font></i>: 
<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"><b>proof </b></a><div class="add">

<b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_2"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:16_1_1_1_2"/><b>then </b>
(  <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> &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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E3:16_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:16_1_1_1"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_3"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_3"/><b>then </b>
(  <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="c8">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b'</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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:16_1_1_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:16_1_1_1_3"/><b>then </b>
(  <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="c10">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E5:16_1_1_1_3"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
( <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">c'</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<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="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:16_1_1_1_3"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E7:16_1_1_1_3"/><b>then </b>
 <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">c'</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E8:16_1_1_1_3"/><b>then </b><i><font color="Green" title="E19">A13</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E9:16_1_1_1_3"/><b>then </b>
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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E10:16_1_1_1_3"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:16_1_1_1_3"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E11:16_1_1_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E12:16_1_1_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E13:16_1_1_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E14:16_1_1_1_3"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E15:16_1_1_1_3"/><b>then </b>
( <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> &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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:16_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:16_1_1_1_3"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E16:16_1_1_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:16_1_1_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="#E8:16_1_1_1_3"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E17:16_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E8:16_1_1_1_3"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E6:16_1_1_1"/><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_4"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_4"/><b>then </b>
(  <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="c6">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_1_1_1_4"/><b>then </b><i><font color="Green" title="E19">A16</font></i>: 
( <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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:16_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="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="c10">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:16_1_1_1_4"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E5:16_1_1_1_4"/><b>then </b>
( <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">c'</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E6:16_1_1_1_4"/><b>then </b><i><font color="Green" title="E20">A17</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E3:16_1_1_1_4"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E7:16_1_1_1_4"/><b>then </b>
( <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> &amp; <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E3:16_1_1_1_4"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E8:16_1_1_1_4"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E9:16_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E6:16_1_1_1_4"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:16_1_1_1"/><i><font color="Green" title="E18">A18</font></i>: 
 <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_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_5"/>
 <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_5"/><b>then </b>
(  <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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:16_1_1_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:16_1_1_1"/><i><font color="Green" title="E19">A19</font></i>: 
 <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">a'</font>,<font color="Maroon" title="c10">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_6"/>
 <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">a'</font>,<font color="Maroon" title="c10">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_6"/><b>then </b>
(  <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">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="c5">a</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> )
 
<b>by </b><i><a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:16_1_1_1_6"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E9:16_1_1_1"/><i><font color="Green" title="E20">A20</font></i>: 
 <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">a'</font>,<font color="Maroon" title="c10">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_7"><b>proof </b></a><div class="add">

<b>assume </b><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="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">a'</font>,<font color="Maroon" title="c10">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_7"/><b>then </b>
(  <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">a'</font>,<font color="Maroon" title="c10">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="c6">a'</font>,<font color="Maroon" title="c8">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="c6">a'</font>,<font color="Maroon" title="c6">a'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:16_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E4:16_1_1_1_7"/><b>then </b>
 <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">a'</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E5:16_1_1_1_7"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E10:16_1_1_1"/><i><font color="Green" title="E21">A21</font></i>: 
( <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">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> &amp; <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:16_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E11:16_1_1_1"/><b>then </b><i><font color="Green" title="E22">A22</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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;  <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#R4" title="AFF_1:pred.4">//</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> )
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:16_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></i>;<br/><a NAME="E12:16_1_1_1"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="#E6:16_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></i>;<br/><a NAME="E13:16_1_1_1"/><i><font color="Green" title="E24">A24</font></i>: 
(  <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="c6">a'</font>,<font color="Maroon" title="c8">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="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="c10">c'</font>,<font color="Maroon" title="c8">b'</font> )
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E14:16_1_1_1"/><i><font color="Green" title="E25">A25</font></i>: 
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">a'</font>,<font color="Maroon" title="c5">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_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">a'</font>,<font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_8"/><b>then </b>
(  <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">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="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_1_1_1_8"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:16_1_1_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="#E13:16_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E15:16_1_1_1"/>
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">c'</font>,<font color="Maroon" title="c9">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_9"/><i><font color="Green" title="E26">A26</font></i>: 
 <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">c'</font>,<font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_9"/><b>then </b>
(  <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">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="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_1_1_1_9"/><b>then </b><i><font color="Green" title="E27">A27</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E3:16_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E13:16_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:16_1_1_1_9"/><i><font color="Green" title="E28">A28</font></i>: 
 <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">c'</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_9_1"><b>now </b></a><div class="add"><b>assume </b><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">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:16_1_1_1_9_1"/><b>then </b>
<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c6">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="txt" href="#E10:16_1_1_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E3:16_1_1_1_9_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><a NAME="E4:16_1_1_1_9_1"/><b>then </b>
<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">a'</font>,<font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E5:16_1_1_1_9_1"/><b>then </b>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font> or <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> )
 <b>by </b><i><a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E13:16_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E6:16_1_1_1_9_1"/><b>then </b>
 <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">a'</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E7:16_1_1_1_9_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E8:16_1_1_1_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:16_1_1_1_9"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E13:16_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="#E1:16_1_1_1_9"><i><font color="Green" title="E26">A26</font></i></a>, <a class="txt" href="#E4:16_1_1_1_9"><i><font color="Green" title="E28">A28</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_1_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E6:16_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="#E3:16_1_1_1_9"><i><font color="Green" title="E27">A27</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E16:16_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="#E1:16_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E2:16_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E6:16_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="#E7:16_1_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="#E8:16_1_1_1"><i><font color="Green" title="E19">A19</font></i></a>, <a class="txt" href="#E9:16_1_1_1"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="#E11:16_1_1_1"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="#E12:16_1_1_1"><i><font color="Green" title="E23">A23</font></i></a>, <a class="txt" href="#E13:16_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="#E14:16_1_1_1"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="aff_3.html#D6" target="_self" title="AFF_3:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E17:16_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E10">A29</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">b'</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">c'</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">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="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">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="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">c'</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Olive" title="b4">a'</font>,<font color="Olive" title="b8">c'</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">b'</font>,<font color="Olive" title="b4">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="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">b'</font>,<font color="Olive" title="b8">c'</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">a'</font>,<font color="Olive" title="b8">c'</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">a'</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b'</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">c'</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">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="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">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="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">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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Olive" title="b2">a'</font>,<font color="Olive" title="b6">c'</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">b'</font>,<font color="Olive" title="b2">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="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">b'</font>,<font color="Olive" title="b6">c'</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">a'</font>,<font color="Olive" title="b6">c'</font></span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">b'</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">c'</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:()"><i><font color="Red">::  thesis: </font></i></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">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="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">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="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">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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</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">b'</font>,<font color="Maroon" title="c6">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="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">b'</font>,<font color="Maroon" title="c10">c'</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">a'</font>,<font color="Maroon" title="c10">c'</font> )</span><br/>





















<b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green" title="E11">A30</font></i>: 
( <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">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="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">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="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">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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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">b'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</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">b'</font>,<font color="Maroon" title="c6">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="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">b'</font>,<font color="Maroon" title="c10">c'</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>,<font color="Maroon" title="c10">c'</font></span><br/>

<b>assume </b><a NAME="E2:16_2_1"/><i><font color="Green" title="E12">A31</font></i>: 
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">a'</font>,<font color="Maroon" title="c10">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:16_2_1"/><i><font color="Green" title="E13">A32</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <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> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c8">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">A'</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/>
<b>set </b><font color="Maroon" title="c14">P'</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/>
<b>set </b><font color="Maroon" title="c15">C'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font>;<br/>
<a NAME="E4:16_2_1"/><i><font color="Green" title="E14">A33</font></i>: 
(  <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;  <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;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">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> &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> &amp; <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> &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> &amp; <font color="Maroon" title="c6">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="c6">a'</font>,<font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E3:16_2_1"><i><font color="Green" title="E13">A32</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E5:16_2_1"/><i><font color="Green" title="E15">A34</font></i>: 
( <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E6:16_2_1"/><b>then </b><i><font color="Green" title="E16">A35</font></i>: 
( <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a'</font> &amp; <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">b'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E7:16_2_1"/><i><font color="Green" title="E17">A36</font></i>: 
 <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#R4" title="AFF_1:pred.4">//</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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E3:16_2_1"><i><font color="Green" title="E13">A32</font></i></a>, <a class="txt" href="#E4:16_2_1"><i><font color="Green" title="E14">A33</font></i></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></i>;<br/>
<a NAME="E8:16_2_1"/><i><font color="Green" title="E18">A37</font></i>: 
<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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_1_1"/><i><font color="Green" title="E19">A38</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:16_2_1_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:16_2_1_1"/><b>then </b>
(  <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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E5:16_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E6:16_2_1_1"/><b>then </b>
 <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">a'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E7:16_2_1_1"/><b>then </b><i><font color="Green" title="E20">A39</font></i>: 
( <font color="Maroon" title="c8">b'</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">a'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E8:16_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">c'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E1:16_2_1_1"><i><font color="Green" title="E19">A38</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E9:16_2_1_1"/><b>then </b>
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> or <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E10:16_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E2:16_2_1"><i><font color="Green" title="E12">A31</font></i></a>, <a class="txt" href="#E5:16_2_1"><i><font color="Green" title="E15">A34</font></i></a>, <a class="txt" href="#E7:16_2_1_1"><i><font color="Green" title="E20">A39</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:16_2_1"/><i><font color="Green" title="E19">A40</font></i>: 
<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"><b>proof </b></a><div class="add">

<b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_2"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:16_2_1_2"/><b>then </b>
(  <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> &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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E8:16_2_1"><i><font color="Green" title="E18">A37</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:16_2_1"/><i><font color="Green" title="E20">A41</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_1_3"/><i><font color="Green" title="E21">A42</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_3"/><b>then </b>
(  <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="c8">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_2_1_3"/><b>then </b>
( <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">b'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E9:16_2_1"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:16_2_1_3"/><b>then </b>
(  <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="c10">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E5:16_2_1_3"/><b>then </b>
( <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">c'</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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E6:16_2_1_3"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E2:16_2_1"><i><font color="Green" title="E12">A31</font></i></a>, <a class="txt" href="#E5:16_2_1"><i><font color="Green" title="E15">A34</font></i></a>, <a class="txt" href="#E1:16_2_1_3"><i><font color="Green" title="E21">A42</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E7:16_2_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E8:16_2_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E9:16_2_1_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E10:16_2_1_3"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E11:16_2_1_3"/><b>then </b>
(  <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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E12:16_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E9:16_2_1"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:16_2_1"/><i><font color="Green" title="E21">A43</font></i>: 
 <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_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_1_4"/>
 <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_4"/><b>then </b>
(  <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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E4:16_2_1"><i><font color="Green" title="E14">A33</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:16_2_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E9:16_2_1"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:16_2_1"/><i><font color="Green" title="E22">A44</font></i>: 
 <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">a'</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="#E2:16_2_1"><i><font color="Green" title="E12">A31</font></i></a>, <a class="txt" href="#E4:16_2_1"><i><font color="Green" title="E14">A33</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>
<a NAME="E13:16_2_1"/><i><font color="Green" title="E23">A45</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_5"><b>proof </b></a><div class="add">

<b>assume </b><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">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_5"/><b>then </b>
(  <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="c6">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_2_1_5"/><b>then </b><i><font color="Green" title="E24">A46</font></i>: 
( <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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E10:16_2_1"><i><font color="Green" title="E20">A41</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:16_2_1_5"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E5:16_2_1_5"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:16_2_1_5"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E7:16_2_1_5"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E8:16_2_1_5"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:16_2_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E3:16_2_1_5"><i><font color="Green" title="E24">A46</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:16_2_1"/><i><font color="Green" title="E24">A47</font></i>: 
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">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_1_6"/><i><font color="Green" title="E25">A48</font></i>: 
 <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">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_2_1_6"/><b>then </b>
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E8:16_2_1"><i><font color="Green" title="E18">A37</font></i></a>, <a class="txt" href="#E1:16_2_1_6"><i><font color="Green" title="E25">A48</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:16_2_1_6"/><b>then </b>
(  <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="c8">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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E5:16_2_1"><i><font color="Green" title="E15">A34</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E5:16_2_1_6"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E8:16_2_1"><i><font color="Green" title="E18">A37</font></i></a>, <a class="txt" href="#E13:16_2_1"><i><font color="Green" title="E23">A45</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:16_2_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E5:16_2_1"><i><font color="Green" title="E15">A34</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:16_2_1"/><i><font color="Green" title="E25">A49</font></i>: 
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">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_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="c6">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_1_7"/><b>then </b>
(  <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">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="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> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_2_1_7"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="txt" href="#E9:16_2_1"><i><font color="Green" title="E19">A40</font></i></a>, <a class="txt" href="#E10:16_2_1"><i><font color="Green" title="E20">A41</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:16_2_1"/>
(  <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="c10">c'</font>,<font color="Maroon" title="c8">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="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="c6">a'</font>,<font color="Maroon" title="c8">b'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E11">A30</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E17:16_2_1"/><b>then </b>
 <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#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="#E1:16_2"><i><font color="Green" title="E10">A29</font></i></a>, <a class="txt" href="#E3:16_2_1"><i><font color="Green" title="E13">A32</font></i></a>, <a class="txt" href="#E4:16_2_1"><i><font color="Green" title="E14">A33</font></i></a>, <a class="txt" href="#E6:16_2_1"><i><font color="Green" title="E16">A35</font></i></a>, <a class="txt" href="#E7:16_2_1"><i><font color="Green" title="E17">A36</font></i></a>, <a class="txt" href="#E11:16_2_1"><i><font color="Green" title="E21">A43</font></i></a>, <a class="txt" href="#E12:16_2_1"><i><font color="Green" title="E22">A44</font></i></a>, <a class="txt" href="#E13:16_2_1"><i><font color="Green" title="E23">A45</font></i></a>, <a class="txt" href="#E14:16_2_1"><i><font color="Green" title="E24">A47</font></i></a>, <a class="txt" href="#E15:16_2_1"><i><font color="Green" title="E25">A49</font></i></a>, <a class="ref" href="aff_3.html#D8" target="_self" title="AFF_3:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E18:16_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:16_2_1"><i><font color="Green" title="E12">A31</font></i></a>, <a class="txt" href="#E4:16_2_1"><i><font color="Green" title="E14">A33</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><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> )
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E9">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
