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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> implies <font color="Maroon" title="c1">AP</font> is <a href="aff_3.html#V5" title="AFF_3:attr.5">satisfying_DES2</a> )</span><br/>

<b>assume </b><a NAME="E1:15"/><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>
 ; <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#V5" title="AFF_3:attr.5">satisfying_DES2</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#D5" target="_self" title="AFF_3:def.5">AFF_3:def 5</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; <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a'</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> 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="b9">p</font>,<font color="Olive" title="b10">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">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; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a'</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> 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="b7">p</font>,<font color="Olive" title="b8">q</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; <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> &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> 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="c11">p</font>,<font color="Maroon" title="c12">q</font> )</span><br/>





















<b>assume </b><b>that </b><br/><a NAME="E2:15"/><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="E3:15"/><i><font color="Green" title="E11">A3</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green" title="E12">A4</font></i>: 
( <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; <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> )
 <b>and </b><br/><a NAME="E5:15"/><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="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> )
 ; <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="c11">p</font>,<font color="Maroon" title="c12">q</font></span><br/>

<a NAME="E6:15"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c3">P</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="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green" title="E15">A7</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="#E4:15"><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/>
<b>set </b><font color="Maroon" title="c13">K</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">N</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/>
<b>set </b><font color="Maroon" title="c15">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>;<br/>
<b>set </b><font color="Maroon" title="c16">T</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="E8:15"/><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="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="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;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</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="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="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="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> &amp; <font color="Maroon" title="c7">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="c7">b</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="c7">b</font>,<font color="Maroon" title="c9">c</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="c7">b</font>,<font color="Maroon" title="c9">c</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">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="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E7:15"><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="E9:15"/><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="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="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E7:15"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></i>;<br/>
<a NAME="E10:15"/><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="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E11:15"/><i><font color="Green" title="E19">A11</font></i>: 
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c9">c</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="E2:15_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="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>
 
<b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E9:15"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E10:15"><i><font color="Green" title="E18">A10</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="E12:15"/><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="c7">b</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="c8">b'</font>,<font color="Maroon" title="c10">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</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="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="E2:15_2"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E11:15"><i><font color="Green" title="E19">A11</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="E3:15_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E6:15"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</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>
<b>set </b><font color="Maroon" title="c17">A'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>;<br/>
<b>set </b><font color="Maroon" title="c18">P'</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/>
<a NAME="E13:15"/><i><font color="Green" title="E21">A13</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">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="c6">a'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c7">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="c7">b</font>,<font color="Maroon" title="c5">a</font> &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="c7">b</font>,<font color="Maroon" title="c5">a</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="c7">b</font>,<font color="Maroon" title="c5">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="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="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="c8">b'</font>,<font color="Maroon" title="c6">a'</font> )
 
<b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E7:15"><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="E14:15"/><i><font color="Green" title="E22">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="15_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3"/><i><font color="Green" title="E23">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="E2:15_3"/><b>then </b><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E12:15"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E3:15_3"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">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="15_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">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="E2:15_3_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</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="E3:15_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</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>
<b>hence </b><a NAME="E4:15_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E1:15_3"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E2:15_3"><i><font color="Green" title="E24">A16</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>
<b>consider </b><font color="Maroon" title="c19">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E16:15"/><i><font color="Green" title="E23">A17</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">M</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> <font color="Maroon" title="c19">M</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E17:15"/><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c19">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 
<b>by </b><i><a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E18:15"/><i><font color="Green" title="E25">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="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c19">M</font>
 
<b>by </b><i><a class="txt" href="#E9:15"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E19:15"/><i><font color="Green" title="E26">A20</font></i>: 
(  not <font color="Maroon" title="c19">M</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="c7">b</font>,<font color="Maroon" title="c9">c</font> or  not  <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="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c19">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_4"/>
( <font color="Maroon" title="c19">M</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="c7">b</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="c8">b'</font>,<font color="Maroon" title="c10">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c19">M</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="E2:15_4"/><b>then </b>
 <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="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="c7">b</font>,<font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:15_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E12:15"><i><font color="Green" title="E20">A12</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>
<div><i><font color="Green" title="E27">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_5"/>
not <font color="Maroon" title="c19">M</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="c7">b</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"> <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></span><br/><b>then consider </b><font color="Maroon" title="c20">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E3:15_5"/><i><font color="Green" title="E28">A22</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">M</font> &amp; <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E17:15"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/><a NAME="E4:15_5"/>
not <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_5_1"/><i><font color="Green" title="E29">A23</font></i>: 
<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</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"> contradiction</span><br/>

<a NAME="E2:15_5_1"/>
<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="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:15_5_1"/><b>then </b>
<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="#E1:15_5_1"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E4:15_5_1"/><b>then </b>
<font color="Maroon" title="c20">x</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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<a NAME="E5:15_5_1"/><b>then </b>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E3:15_5"><i><font color="Green" title="E28">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="E6:15_5_1"/><b>then </b><i><font color="Green" title="E30">A24</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="c19">M</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="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E6:15"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E17:15"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="#E3:15_5"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_5_1_1"/>
<font color="Maroon" title="c7">b</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="E2:15_5_1_1"/><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="c8">b'</font>,<font color="Maroon" title="c6">a'</font>
 <b>by </b><i><a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E3:15_5_1_1"/><b>then </b>
<font color="Maroon" title="c6">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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E14:15"><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="E4:15_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</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="E8:15_5_1"/><b>then </b>
<font color="Maroon" title="c7">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="c5">a</font>,<font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E18:15"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="#E6:15_5_1"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E9:15_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><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><b>then consider </b><font color="Maroon" title="c21">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E6:15_5"/><i><font color="Green" title="E29">A25</font></i>: 
( <font color="Maroon" title="c21">y</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="c8">b'</font>,<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c21">y</font> )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></i>;<br/><a NAME="E7:15_5"/><i><font color="Green" title="E30">A26</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="c21">y</font>,<font color="Maroon" title="c20">x</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="c20">x</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="c20">x</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E18:15"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="#E3:15_5"><i><font color="Green" title="E28">A22</font></i></a>, <a class="txt" href="#E6:15_5"><i><font color="Green" title="E29">A25</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>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E8:15_5"/><i><font color="Green" title="E31">A27</font></i>: 
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="c21">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_5_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="c21">y</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="E2:15_5_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c21">y</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="c8">b'</font>,<font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c8">b'</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="E3:15_5_2"/><b>then </b>
( <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c20">x</font> )
 
<b>by </b><i><a class="txt" href="#E7:15_5"><i><font color="Green" title="E30">A26</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E4:15_5_2"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c6">a'</font>
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E6:15"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E6:15_5"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E5:15_5_2"/><b>then </b><i><font color="Green" title="E32">A28</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> or  <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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">M</font> )
 
<b>by </b><i><a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E17:15"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="#E3:15_5"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_5_2_1"/>
<font color="Maroon" title="c20">x</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="E2:15_5_2_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="c7">b</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</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="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E3:15_5"><i><font color="Green" title="E28">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="E3:15_5_2_1"/><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="c8">b'</font>,<font color="Maroon" title="c6">a'</font> or <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="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</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#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E4:15_5_2_1"/><b>then </b>
<font color="Maroon" title="c6">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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E14:15"><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>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/><b>hence </b><a NAME="E5:15_5_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</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="E7:15_5_2"/><b>then </b>
 <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="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>
 
<b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E5:15_5_2"><i><font color="Green" title="E32">A28</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E8:15_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</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="E9:15_5"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_5_3"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</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="E2:15_5_3"/><b>then </b>
( <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  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</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="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E3:15_5"><i><font color="Green" title="E28">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="E3:15_5_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="c8">b'</font>,<font color="Maroon" title="c6">a'</font> or <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="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</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#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E4:15_5_3"/><b>then </b>
<font color="Maroon" title="c6">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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E14:15"><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>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:15_5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</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="E10:15_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="c6">a'</font>,<font color="Maroon" title="c21">y</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E6:15_5"><i><font color="Green" title="E29">A25</font></i></a>, <a class="txt" href="#E7:15_5"><i><font color="Green" title="E30">A26</font></i></a>, <a class="txt" href="#E8:15_5"><i><font color="Green" title="E31">A27</font></i></a>, <a class="ref" href="aff_3.html#D6" target="_self" title="AFF_3:def.6">Def6</a></i>;<br/><a NAME="E11:15_5"/><b>then </b>
<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c21">y</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>
 <b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E7:15"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E12:15_5"/><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="c21">y</font>,<font color="Maroon" title="c10">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="E13:15_5"/><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="c21">y</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="E14:15_5"/><b>then </b>
<font color="Maroon" title="c21">y</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="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/><a NAME="E15:15_5"/><b>then </b>
(  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E6:15_5"><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="E16:15_5"/><b>then </b>
( <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> or  <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="c20">x</font> )
 <b>by </b><i><a class="txt" href="#E6:15_5"><i><font color="Green" title="E29">A25</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#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/><a NAME="E17:15_5"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">c'</font>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/><a NAME="E18:15_5"/><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="c19">M</font> or <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="c7">b</font>,<font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E3:15_5"><i><font color="Green" title="E28">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="E19:15_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="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="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E18:15"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E20:15_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="c12">q</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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E11:15"><i><font color="Green" title="E19">A11</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="E21:15_5"/>
<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>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E6:15"><i><font color="Green" title="E14">A6</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_6"/>
not  <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="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c19">M</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="c11">p</font>,<font color="Maroon" title="c12">q</font></span><br/><b>then consider </b><font color="Maroon" title="c20">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E3:15_6"/><i><font color="Green" title="E28">A29</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">M</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E17:15"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/><a NAME="E4:15_6"/>
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_6_1"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</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"> contradiction</span><br/>

<a NAME="E2:15_6_1"/><b>then </b>
( <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">C</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="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:15_6_1"/><b>then </b>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E4:15_6_1"/><b>then </b>
<font color="Maroon" title="c20">x</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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<a NAME="E5:15_6_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="c19">M</font> or <font color="Maroon" title="c10">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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E3:15_6"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E6:15_6_1"/><b>then </b>
( <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> or <font color="Maroon" title="c19">M</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> )
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E6:15"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E17:15"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E7:15_6_1"/><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="c8">b'</font>,<font color="Maroon" title="c5">a</font> or  <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="c8">b'</font>,<font color="Maroon" title="c6">a'</font> )
 
<b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">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="E8:15_6_1"/><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  <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="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> )
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E14:15"><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>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E9:15_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</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>then consider </b><font color="Maroon" title="c21">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E6:15_6"/><i><font color="Green" title="E29">A30</font></i>: 
( <font color="Maroon" title="c21">y</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="c7">b</font>,<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c21">y</font> )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></i>;<br/><a NAME="E7:15_6"/><i><font color="Green" title="E30">A31</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c20">x</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="c20">x</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="c11">p</font>,<font color="Maroon" title="c20">x</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E3:15_6"><i><font color="Green" title="E28">A29</font></i></a>, <a class="txt" href="#E6:15_6"><i><font color="Green" title="E29">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#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E8:15_6"/><i><font color="Green" title="E31">A32</font></i>: 
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="c21">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_6_2"/>
 <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="c21">y</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="E2:15_6_2"/><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="c21">y</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="c21">y</font>,<font color="Maroon" title="c7">b</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="E3:15_6_2"/><b>then </b>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> )
 
<b>by </b><i><a class="txt" href="#E7:15_6"><i><font color="Green" title="E30">A31</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E4:15_6_2"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E6:15"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E6:15_6"><i><font color="Green" title="E29">A30</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E5:15_6_2"/><b>then </b><i><font color="Green" title="E32">A33</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">M</font> )
 
<b>by </b><i><a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E17:15"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="#E3:15_6"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_6_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_6_2_1"/>
<font color="Maroon" title="c20">x</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="E2:15_6_2_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="c8">b'</font> or  <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#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> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E3:15_6"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E3:15_6_2_1"/><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="c8">b'</font>,<font color="Maroon" title="c5">a</font> or <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="c6">a'</font> )
 <b>by </b><i><a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</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#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E4:15_6_2_1"/><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>
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E14:15"><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>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/><b>hence </b><a NAME="E5:15_6_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</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="E7:15_6_2"/><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="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="c7">b</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E18:15"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="#E5:15_6_2"><i><font color="Green" title="E32">A33</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E8:15_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</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="E9:15_6"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_6_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_6_3"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</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="E2:15_6_3"/><b>then </b>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b'</font> or  <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#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> )
 
<b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E3:15_6"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E3:15_6_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="c8">b'</font>,<font color="Maroon" title="c5">a</font> or <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="c6">a'</font> )
 
<b>by </b><i><a class="txt" href="#E13:15"><i><font color="Green" title="E21">A13</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#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E4:15_6_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>
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E14:15"><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>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:15_6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</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="E10:15_6"/><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="c5">a</font>,<font color="Maroon" title="c21">y</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E6:15_6"><i><font color="Green" title="E29">A30</font></i></a>, <a class="txt" href="#E7:15_6"><i><font color="Green" title="E30">A31</font></i></a>, <a class="txt" href="#E8:15_6"><i><font color="Green" title="E31">A32</font></i></a>, <a class="ref" href="aff_3.html#D6" target="_self" title="AFF_3:def.6">Def6</a></i>;<br/><a NAME="E11:15_6"/><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="c21">y</font>
 <b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E7:15"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E12:15_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="c9">c</font>,<font color="Maroon" title="c21">y</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="E13:15_6"/><b>then </b>
<font color="Maroon" title="c21">y</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>
 <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="E14:15_6"/><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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E6:15_6"><i><font color="Green" title="E29">A30</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E15:15_6"/><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> or  <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="c20">x</font> )
 <b>by </b><i><a class="txt" href="#E6:15_6"><i><font color="Green" title="E29">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#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/><a NAME="E16:15_6"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/><a NAME="E17:15_6"/><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="c19">M</font> or <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="c7">b</font>,<font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E3:15_6"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E18:15_6"/><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="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="txt" href="#E8:15"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E16:15"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E18:15"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E19:15_6"/><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="c12">q</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="#E2:15"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E11:15"><i><font color="Green" title="E19">A11</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="E20:15_6"/>
<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>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E6:15"><i><font color="Green" title="E14">A6</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>
<b>hence </b><a NAME="E22:15"/>
<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>
 <b>by </b><i><a class="txt" href="#E19:15"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="#E20:15"><i><font color="Green" title="E27">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>
