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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> <a href="aff_3.html#NR2" title="AFF_3:NR.2">satisfies_DES1_1</a>  implies <font color="Maroon" title="c1">AP</font> <a href="aff_3.html#NR1" title="AFF_3:NR.1">satisfies_DES1</a>  )</span><br/>

<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_3.html#NR2" title="AFF_3:NR.2">satisfies_DES1_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> <a href="aff_3.html#NR1" title="AFF_3:NR.1">satisfies_DES1</a> </span><br/>

<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#D1" target="_self" title="AFF_3:def.1">AFF_3:def 1</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">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">a'</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">b'</font>, <font color="Olive" title="b8">c</font>, <font color="Olive" title="b9">c'</font>, <font color="Olive" title="b10">p</font>, <font color="Olive" title="b11">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="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</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="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="b3">o</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">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">o</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">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="b3">o</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="Olive" title="b9">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="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">c</font> &amp; <font color="Olive" title="b10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b11">q</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; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b7">b'</font>,<font color="Olive" title="b5">a'</font>,<font color="Olive" title="b9">c'</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a'</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="b10">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b7">b'</font>,<font color="Olive" title="b5">a'</font>,<font color="Olive" title="b10">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="b8">c</font>,<font color="Olive" title="b11">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b7">b'</font>,<font color="Olive" title="b9">c'</font>,<font color="Olive" title="b11">q</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a'</font>,<font color="Olive" title="b9">c'</font> holds <br/><font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b10">p</font>,<font color="Olive" title="b11">q</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">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">b'</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">c'</font>, <font color="Olive" title="b8">p</font>, <font color="Olive" title="b9">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="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</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="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="Olive" title="b1">o</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">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="b1">o</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">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="b1">o</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="Olive" title="b7">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="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">q</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; 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; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a'</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="b8">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="b3">a'</font>,<font color="Olive" title="b8">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="b6">c</font>,<font color="Olive" title="b9">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="b7">c'</font>,<font color="Olive" title="b9">q</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b7">c'</font> holds <br/><font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">p</font>,<font color="Olive" title="b9">q</font></span><br/><b>let </b><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">c'</font>, <font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">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="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</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="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="c5">o</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">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="c5">o</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">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="c5">o</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="c11">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="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</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="c5">o</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="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</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; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">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="c7">a'</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="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c12">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="c10">c</font>,<font color="Maroon" title="c13">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">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="c7">a'</font>,<font color="Maroon" title="c11">c'</font> implies <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="c12">p</font>,<font color="Maroon" title="c13">q</font> )</span><br/>























<b>assume </b><b>that </b><br/><a NAME="E2:18"/><i><font color="Green" title="E10">A2</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> )
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</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="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> )
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c5">o</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">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="c5">o</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">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="c5">o</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="c11">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="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</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="c5">o</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="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</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; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">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="c7">a'</font> )
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green" title="E13">A5</font></i>: 
(  <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="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c12">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="c10">c</font>,<font color="Maroon" title="c13">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">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="c7">a'</font>,<font color="Maroon" title="c11">c'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c12">p</font>,<font color="Maroon" title="c13">q</font></span><br/>

<b>set </b><font color="Maroon" title="c14">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>;<br/>
<b>set </b><font color="Maroon" title="c15">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>;<br/>
<a NAME="E6:18"/>
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</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> &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> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</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="c8">b</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="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="c8">b</font>,<font color="Maroon" title="c6">a</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="c8">b</font>,<font color="Maroon" title="c6">a</font> &amp; <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="c8">b</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="c8">b</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c12">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="c8">b</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c13">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="c8">b</font>,<font color="Maroon" title="c10">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:18"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green" title="E15">A7</font></i>: 
( <font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green" title="E16">A8</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E10:18"/><i><font color="Green" title="E17">A9</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E11:18"/><i><font color="Green" title="E18">A10</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E12:18"/><i><font color="Green" title="E19">A11</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">o</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="c6">a</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="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E13:18"/><i><font color="Green" title="E20">A12</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E14:18"/><i><font color="Green" title="E21">A13</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<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="E2:18_1"/><b>then </b>
<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="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</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="E3:18_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</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="E15:18"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="E2:18_2"/><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="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:18"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E14:18"><i><font color="Green" title="E21">A13</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#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="E3:18_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</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><i><font color="Green" title="E23">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3"/><i><font color="Green" title="E24">A16</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">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="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font></span><br/><a NAME="E2:18_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:18"><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="E3:18_3"/>
<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="c12">p</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E1:18"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:18"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E8:18"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E11:18"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E12:18"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="aff_3.html#E13:18"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E14:18"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="aff_3.html#E15:18"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="aff_3.html#E1:18_3"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="aff_3.html#D2" target="_self" title="AFF_3:def.2">Def2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_4"/><i><font color="Green" title="E24">A17</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">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="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font></span><br/><b>set </b><font color="Maroon" title="c16">A'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>;<br/><b>set </b><font color="Maroon" title="c17">C'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>;<br/><a NAME="E2:18_4"/><i><font color="Green" title="E25">A18</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E1:18_4"><i><font color="Green" title="E24">A17</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E3:18_4"/><i><font color="Green" title="E26">A19</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</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="c9">b'</font>,<font color="Maroon" title="c11">c'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c9">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="c9">b'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c7">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="c9">b'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c12">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="c9">b'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c9">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="c9">b'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c11">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="c9">b'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c13">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="c9">b'</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E5:18"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E8:18"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E4:18_4"/><b>then </b><i><font color="Green" title="E27">A20</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><div><i><font color="Green" title="E28">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_4_1"/>
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c12">p</font>,<font color="Maroon" title="c13">q</font></span><br/><a NAME="E2:18_4_1"/><b>then </b><i><font color="Green" title="E29">A22</font></i>: 
<font color="Maroon" title="c13">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="c9">b'</font>,<font color="Maroon" title="c7">a'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E1:18_4"><i><font color="Green" title="E24">A17</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E3:18_4_1"/><b>then </b><i><font color="Green" title="E30">A23</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E4:18_4_1"/>
 <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="c9">b'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E5:18"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="aff_3.html#E2:18_4_1"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E5:18_4_1"/><b>then </b>
<font color="Maroon" title="c9">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="c8">b</font>,<font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/><a NAME="E6:18_4_1"/><b>then </b><i><font color="Green" title="E31">A24</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E10:18"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E7:18_4_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_4_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</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="E2:18_4_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="c7">a'</font>,<font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:18_4_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>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</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="E4:18_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</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="E8:18_4_1"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4_1"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="aff_3.html#E6:18_4_1"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E9:18_4_1"/>
<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="c12">p</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_4_2"/>
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">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="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font></span><br/><a NAME="E2:18_4_2"/><b>then </b><i><font color="Green" title="E29">A25</font></i>: 
<font color="Maroon" title="c12">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="c9">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18_4"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E3:18_4_2"/><b>then </b><i><font color="Green" title="E30">A26</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E4:18_4_2"/>
 <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="c9">b'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E5:18"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="aff_3.html#E2:18_4_2"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E5:18_4_2"/><b>then </b>
<font color="Maroon" title="c9">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="c8">b</font>,<font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/><a NAME="E6:18_4_2"/><b>then </b><i><font color="Green" title="E31">A27</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E9:18"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E7:18_4_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">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="c8">b</font>,<font color="Maroon" title="c10">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_4_2_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">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="c8">b</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="E2:18_4_2_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="c11">c'</font>,<font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:18_4_2_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="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E15:18"><i><font color="Green" title="E22">A14</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="E4:18_4_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</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="E8:18_4_2"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4_2"><i><font color="Green" title="E30">A26</font></i></a>, <a class="txt" href="aff_3.html#E6:18_4_2"><i><font color="Green" title="E31">A27</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E9:18_4_2"/>
<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="c12">p</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</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="E7:18_4"/>
<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="c12">p</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:18_4"><i><font color="Green" title="E28">A21</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><b>end;</b></div>
<b>hence </b><a NAME="E18:18"/>
<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="c12">p</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E16:18"><i><font color="Green" title="E23">A15</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>
