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<span class="kw">let </span><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V4" title="AFF_3:attr.4">satisfying_DES1_3</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V4" title="AFF_3:attr.4">satisfying_DES1_3</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a> </span><br/>

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





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E4:26"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E5:26"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E6:26"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E7:26"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E8:26"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E9:26"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E10:26"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E11:26"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E12:26"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E13:26"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E14:26"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E15:26"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E16:26"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">and </span><br/><a NAME="E17:26"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E18:26"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E19:26"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E20:26"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E21:26"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E22:26"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E23:26"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font></span><br/>

<a NAME="E24:26"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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">a9</font>,<font color="Maroon" title="c10">c9</font>;<br/>
<a NAME="E25:26"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E26:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A26</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E27:26"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
( <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E29">A21</font></span></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#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E28:26"/><span class="lab"><font color="Green" title="E36">A28</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E29:26"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E30:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A29</font></span>: 
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E31:26"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">b9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E32:26"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">assume </span><a NAME="E33:26"/><span class="lab"><font color="Green" title="E40">A32</font></span>: 
not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E34:26"/><span class="lab"><font color="Green" title="E41">A33</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E42">A34</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:26_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E4:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A35</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E5:26_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E27">A19</font></span></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></span>;<br/>
<a NAME="E6:26_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E4:26_1"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<a NAME="E7:26_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E8:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A36</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> or <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E9:26_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E10:26_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font> or <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:26_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E8:26_1"><span class="lab"><font color="Green" title="E44">A36</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:26"/><span class="lab"><font color="Green" title="E42">A37</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E43">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_2_1"/><span class="lab"><font color="Green" title="E44">A39</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_2_1"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E3:26_2_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:26_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E1:26_2_1"><span class="lab"><font color="Green" title="E44">A39</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E5:26_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E1:26_2_1"><span class="lab"><font color="Green" title="E44">A39</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E6:26_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:26_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:26_2"/><span class="lab"><font color="Green" title="E44">A40</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E29">A21</font></span></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></span>;<br/>
<span class="kw">assume </span><a NAME="E3:26_2"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:26_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E5:26_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font> or <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E34:26"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="txt" href="#E2:26_2"><span class="lab"><font color="Green" title="E44">A40</font></span></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></span>;<br/>
<a NAME="E6:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A42</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E7:26_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E27">A19</font></span></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></span>;<br/>
<a NAME="E8:26_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E6:26_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E1:26_2"><span class="lab"><font color="Green" title="E43">A38</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E9:26_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E3:26_2"><span class="lab"><font color="Green" title="E45">A41</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E36:26"/><span class="lab"><font color="Green" title="E43">A43</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A44</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E3:26_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E29">A21</font></span></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></span>;<br/>
<a NAME="E4:26_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A45</font></span>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E35:26"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E2:26_3"><span class="lab"><font color="Green" title="E44">A44</font></span></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></span>;<br/>
<a NAME="E5:26_3"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E6:26_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E7:26_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E4:26_3"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E8:26_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E4:26_3"><span class="lab"><font color="Green" title="E45">A45</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E9:26_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E37:26"/><span class="lab"><font color="Green" title="E44">A46</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_4"/><span class="lab"><font color="Green" title="E45">A47</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_4"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E29">A21</font></span></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></span>;<br/>
<a NAME="E3:26_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E34:26"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="txt" href="#E35:26"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E1:26_4"><span class="lab"><font color="Green" title="E45">A47</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E4:26_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E38:26"/><span class="lab"><font color="Green" title="E45">A48</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E39:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A49</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E40:26"/><span class="lab"><font color="Green" title="E47">A50</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_5"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:26_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A51</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E4:26_5"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E29">A21</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E5:26_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E34:26"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E48">A51</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E41:26"/><span class="lab"><font color="Green" title="E48">A52</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_6"/><span class="lab"><font color="Green" title="E49">A53</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_6"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E27">A19</font></span></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></span>;<br/>
<a NAME="E3:26_6"/><span class="kw">then </span>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E40:26"><span class="lab"><font color="Green" title="E47">A50</font></span></a>, <a class="txt" href="#E1:26_6"><span class="lab"><font color="Green" title="E49">A53</font></span></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></span>;<br/>
<a NAME="E4:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A54</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E5:26_6"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E27">A19</font></span></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></span>;<br/>
<a NAME="E6:26_6"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E36:26"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="txt" href="#E40:26"><span class="lab"><font color="Green" title="E47">A50</font></span></a>, <a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E50">A54</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E7:26_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E42:26"/><span class="lab"><font color="Green" title="E49">A55</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E43:26"/><span class="lab"><font color="Green" title="E50">A56</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E51">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_7"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_7"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A58</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="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="c9">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:26"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E3:26_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:26_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E34:26"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="txt" href="#E2:26_7"><span class="lab"><font color="Green" title="E52">A58</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E45:26"/><span class="lab"><font color="Green" title="E52">A59</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:26"><span class="lab"><font color="Green" title="E45">A48</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E46:26"/><span class="lab"><font color="Green" title="E53">A60</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:26"><span class="lab"><font color="Green" title="E45">A48</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E48:26"/><span class="lab"><font color="Green" title="E54">A61</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E49:26"/><span class="lab"><font color="Green" title="E55">A62</font></span>: 
<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">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E39:26"><span class="lab"><font color="Green" title="E46">A49</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E45:26"><span class="lab"><font color="Green" title="E52">A59</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E56">A63</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_8"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_8"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_8"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E35:26"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E39:26"><span class="lab"><font color="Green" title="E46">A49</font></span></a>, <a class="txt" href="#E45:26"><span class="lab"><font color="Green" title="E52">A59</font></span></a>, <a class="txt" href="#E49:26"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E46:26"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E51:26"/><span class="lab"><font color="Green" title="E57">A64</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E52:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A65</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E43:26"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E59">A66</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_9"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_9"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_9"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E35:26"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E48:26"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E54:26"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E46:26"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E45:26"><span class="lab"><font color="Green" title="E52">A59</font></span></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></span>;<br/>
<a NAME="E55:26"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E39:26"><span class="lab"><font color="Green" title="E46">A49</font></span></a>, <a class="txt" href="#E51:26"><span class="lab"><font color="Green" title="E57">A64</font></span></a>, <a class="txt" href="#E43:26"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E46:26"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E45:26"><span class="lab"><font color="Green" title="E52">A59</font></span></a>, <a class="txt" href="#E48:26"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="txt" href="#E49:26"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="txt" href="#E30:26"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E27:26"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E31:26"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E28:26"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E36:26"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="txt" href="#E50:26"><span class="lab"><font color="Green" title="E56">A63</font></span></a>, <a class="txt" href="#E53:26"><span class="lab"><font color="Green" title="E59">A66</font></span></a>, <a class="txt" href="#E37:26"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E41:26"><span class="lab"><font color="Green" title="E48">A52</font></span></a>, <a class="ref" href="aff_3.html#D4" target="_self" title="AFF_3:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E56:26"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:26"><span class="lab"><font color="Green" title="E58">A65</font></span></a>, <a class="txt" href="#E48:26"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="txt" href="#E44:26"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
