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<b>let </b><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> <a href="aff_3.html#NR4" title="AFF_3:NR.4">satisfies_DES1_3</a>  implies <font color="Maroon" title="c1">AP</font> <a href="aff_3.html#NR6" title="AFF_3:NR.6">satisfies_DES2_1</a>  )</span><br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_3.html#NR4" title="AFF_3:NR.4">satisfies_DES1_3</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">AP</font> <a href="aff_3.html#NR6" title="AFF_3:NR.6">satisfies_DES2_1</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#D6" target="_self" title="AFF_3:def.6">AFF_3:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">a'</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">b'</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">c'</font>, <font color="Olive" title="b9">p</font>, <font color="Olive" title="b10">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b1">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b2">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b8">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">C</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b'</font>,<font color="Olive" title="b4">a'</font>,<font color="Olive" title="b8">c'</font> &amp; <font color="Olive" title="b9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b10">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b9">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b'</font>,<font color="Olive" title="b4">a'</font>,<font color="Olive" title="b9">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b7">c</font>,<font color="Olive" title="b10">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b'</font>,<font color="Olive" title="b8">c'</font>,<font color="Olive" title="b10">q</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font> holds <br/><font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a'</font>,<font color="Olive" title="b8">c'</font></span><br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">a'</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b'</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">c'</font>, <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b4">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b6">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b'</font>,<font color="Olive" title="b2">a'</font>,<font color="Olive" title="b6">c'</font> &amp; <font color="Olive" title="b7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b'</font>,<font color="Olive" title="b2">a'</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b8">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b'</font>,<font color="Olive" title="b6">c'</font>,<font color="Olive" title="b8">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b6">c'</font></span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">b'</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">c'</font>, <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c11">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font> )</span><br/>





















<b>assume </b><b>that </b><br/><a NAME="E2:26"/><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>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green" title="E12">A4</font></i>: 
<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="E5:26"/><i><font color="Green" title="E13">A5</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>
 <b>and </b><br/><a NAME="E6:26"/><i><font color="Green" title="E14">A6</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="c4">C</font>
 <b>and </b><br/><a NAME="E7:26"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <b>and </b><br/><a NAME="E8:26"/><i><font color="Green" title="E16">A8</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>
 <b>and </b><br/><a NAME="E9:26"/><i><font color="Green" title="E17">A9</font></i>: 
<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>
 <b>and </b><br/><a NAME="E10:26"/><i><font color="Green" title="E18">A10</font></i>: 
<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>
 <b>and </b><br/><a NAME="E11:26"/><i><font color="Green" title="E19">A11</font></i>: 
<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>
 <b>and </b><br/><a NAME="E12:26"/><i><font color="Green" title="E20">A12</font></i>: 
<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>
 <b>and </b><br/><a NAME="E13:26"/><i><font color="Green" title="E21">A13</font></i>: 
<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>
 <b>and </b><br/><a NAME="E14:26"/><i><font color="Green" title="E22">A14</font></i>: 
<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>
 <b>and </b><br/><a NAME="E15:26"/><i><font color="Green" title="E23">A15</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 <b>and </b><br/><a NAME="E16:26"/><i><font color="Green" title="E24">A16</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 <b>and </b><br/><a NAME="E17:26"/><i><font color="Green" title="E25">A17</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="c10">c'</font>
 <b>and </b><br/><a NAME="E18:26"/><i><font color="Green" title="E26">A18</font></i>: 
<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>
 <b>and </b><br/><a NAME="E19:26"/><i><font color="Green" title="E27">A19</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>
 <b>and </b><br/><a NAME="E20:26"/><i><font color="Green" title="E28">A20</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c11">p</font>
 <b>and </b><br/><a NAME="E21:26"/><i><font color="Green" title="E29">A21</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="c9">c</font>,<font color="Maroon" title="c12">q</font>
 <b>and </b><br/><a NAME="E22:26"/><i><font color="Green" title="E30">A22</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="c10">c'</font>,<font color="Maroon" title="c12">q</font>
 <b>and </b><br/><a NAME="E23:26"/><i><font color="Green" title="E31">A23</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font></span><br/>

<b>assume </b><a NAME="E24:26"/><i><font color="Green" title="E32">A24</font></i>: 
not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E25:26"/><i><font color="Green" title="E33">A25</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="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="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="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="aff_3.html#E16:26"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="aff_3.html#E17:26"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E26:26"/><i><font color="Green" title="E34">A26</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="aff_3.html#E14:26"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="aff_3.html#E15:26"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E27:26"/><i><font color="Green" title="E35">A27</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="c12">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E36">A28</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E22:26"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E3:26_1"/><b>then </b><i><font color="Green" title="E37">A29</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> or <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:26"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E12:26"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E13:26"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:26_1"/>
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E23:26"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="aff_3.html#E1:26_1"><i><font color="Green" title="E36">A28</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E5:26_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:26_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E19:26"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E7:26_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="c5">a</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E16:26"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E8:26_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E20:26"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E9:26_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font> or <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:26"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E8:26"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="aff_3.html#E9:26"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E10:26_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E5:26"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:26"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E11:26"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="aff_3.html#E14:26"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="aff_3.html#E24:26"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="aff_3.html#E26:26"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="aff_3.html#E3:26_1"><i><font color="Green" title="E37">A29</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E28:26"/><i><font color="Green" title="E36">A30</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="26_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green" title="E37">A31</font></i>: 
<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="E2:26_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c8">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E21:26"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="aff_3.html#E22:26"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:26_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="c8">b'</font> or <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E3:26"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E11:26"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="aff_3.html#E27:26"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:26_2"/><b>then </b><i><font color="Green" title="E38">A32</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:26"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E12:26"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E19:26"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="aff_3.html#E20:26"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="aff_3.html#E26:26"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2_1"/><i><font color="Green" title="E39">A33</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_2_1"/><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="c7">b</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="c7">b</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E21:26"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="aff_3.html#E23:26"><i><font color="Green" title="E31">A23</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E3:26_2_1"/><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="c7">b</font>,<font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c12">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><a NAME="E4:26_2_1"/><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="c7">b</font>,<font color="Maroon" title="c9">c</font> or <font color="Maroon" title="c7">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="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E5:26_2_1"/><b>then </b>
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E18:26"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="aff_3.html#E1:26_2_1"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E6:26_2_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E7:26_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E16:26"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:26_2"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:26"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E8:26"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="aff_3.html#E9:26"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E4:26_2"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E7:26_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E5:26"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E14:26"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="aff_3.html#E24:26"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="aff_3.html#E1:26_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<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="c11">p</font>,<font color="Maroon" title="c12">q</font>;<br/>
<b>set </b><font color="Maroon" title="c14">M</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="c15">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/>
<a NAME="E29:26"/><i><font color="Green" title="E37">A34</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> 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="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; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="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> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E18:26"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="aff_3.html#E25:26"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E30:26"/><b>then </b><i><font color="Green" title="E38">A35</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E18:26"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="aff_3.html#E23:26"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="aff_3.html#E25:26"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></i>;<br/>
<a NAME="E31:26"/>
not  <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="aff_3.html#E24:26"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">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="E33:26"/><i><font color="Green" title="E39">A36</font></i>: 
( <font color="Maroon" title="c16">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="c5">a</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c16">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="c6">a'</font>,<font color="Maroon" title="c10">c'</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a>, <a class="txt" href="aff_3.html#E24:26"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E34:26"/>
( <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> &amp; <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E14:26"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="aff_3.html#E15:26"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E35:26"/><b>then </b><i><font color="Green" title="E40">A37</font></i>: 
( <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E8:26"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="aff_3.html#E9:26"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E11:26"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="aff_3.html#E12:26"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E13:26"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E36:26"/><i><font color="Green" title="E41">A38</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c8">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c8">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E19:26"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="aff_3.html#E20:26"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="aff_3.html#E21:26"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="aff_3.html#E22:26"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E37:26"/><i><font color="Green" title="E42">A39</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="26_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_3"/>
<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="E2:26_3"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E21:26"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="aff_3.html#E22:26"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:26_3"/><b>then </b><i><font color="Green" title="E43">A40</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or <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="aff_3.html#E4:26"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E12:26"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E13:26"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="aff_3.html#E28:26"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:26_3"/><b>then </b>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:26"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E19:26"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="aff_3.html#E23:26"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="aff_3.html#E26:26"><i><font color="Green" title="E34">A26</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#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:26_3"/><b>then </b>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E6:26_3"/><b>then </b>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E7:26_3"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:26"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E18:26"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="aff_3.html#E26:26"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="aff_3.html#E3:26_3"><i><font color="Green" title="E43">A40</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#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:26_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E16:26"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E38:26"/><i><font color="Green" title="E43">A41</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_4"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_4"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E23:26"><i><font color="Green" title="E31">A23</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:26_4"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E21:26"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E4:26_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E16:26"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="aff_3.html#E27:26"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E44">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_5"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_5"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E21:26"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><b>hence </b><a NAME="E3:26_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E16:26"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="aff_3.html#E27:26"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E40:26"/><i><font color="Green" title="E45">A43</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E24:26"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>
<div><i><font color="Green" title="E46">A44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_6"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_6"/><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>
 <b>by </b><i><a class="txt" href="aff_3.html#E4:26"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E12:26"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E13:26"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="aff_3.html#E28:26"><i><font color="Green" title="E36">A30</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="txt" href="aff_3.html#E33:26"><i><font color="Green" title="E39">A36</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="E3:26_6"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E6:26"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E9:26"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E15:26"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E47">A45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_7"/>
<font color="Maroon" title="c16">x</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="E2:26_7"/><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>
 <b>by </b><i><a class="txt" href="aff_3.html#E4:26"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E12:26"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E13:26"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="aff_3.html#E28:26"><i><font color="Green" title="E36">A30</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="txt" href="aff_3.html#E33:26"><i><font color="Green" title="E39">A36</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="E3:26_7"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E6:26"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E8:26"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="aff_3.html#E15:26"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E43:26"/><i><font color="Green" title="E48">A46</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_8"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_8"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E21:26"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:26_8"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:26"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E12:26"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E13:26"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="aff_3.html#E27:26"><i><font color="Green" title="E35">A27</font></i></a>, <a class="txt" href="aff_3.html#E28:26"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:26_8"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E7:26"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E26:26"><i><font color="Green" title="E34">A26</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="E44:26"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_9"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E19:26"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:26_9"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:26"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E8:26"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="aff_3.html#E9:26"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E38:26"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:26_9"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:26"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E14:26"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="aff_3.html#E19:26"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="aff_3.html#E20:26"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E5:26_9"/><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="aff_3.html#E3:26"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E10:26"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E11:26"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="aff_3.html#E37:26"><i><font color="Green" title="E42">A39</font></i></a>, <a class="txt" href="aff_3.html#E38:26"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:26_9"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E5:26"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E8:26"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="aff_3.html#E14:26"><i><font color="Green" title="E22">A14</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="E45:26"/><b>then </b>
<font color="Maroon" title="c16">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="c11">p</font>,<font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E1:26"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_3.html#E18:26"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="aff_3.html#E25:26"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="aff_3.html#E29:26"><i><font color="Green" title="E37">A34</font></i></a>, <a class="txt" href="aff_3.html#E33:26"><i><font color="Green" title="E39">A36</font></i></a>, <a class="txt" href="aff_3.html#E35:26"><i><font color="Green" title="E40">A37</font></i></a>, <a class="txt" href="aff_3.html#E36:26"><i><font color="Green" title="E41">A38</font></i></a>, <a class="txt" href="aff_3.html#E37:26"><i><font color="Green" title="E42">A39</font></i></a>, <a class="txt" href="aff_3.html#E40:26"><i><font color="Green" title="E45">A43</font></i></a>, <a class="txt" href="aff_3.html#E41:26"><i><font color="Green" title="E46">A44</font></i></a>, <a class="txt" href="aff_3.html#E42:26"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="aff_3.html#E43:26"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="aff_3.html#D4" target="_self" title="AFF_3:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E46:26"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E30:26"><i><font color="Green" title="E38">A35</font></i></a>, <a class="txt" href="aff_3.html#E33:26"><i><font color="Green" title="E39">A36</font></i></a>, <a class="txt" href="aff_3.html#E39:26"><i><font color="Green" title="E44">A42</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>
