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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#V6" title="AFF_3:attr.6">satisfying_DES2_1</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V5" title="AFF_3:attr.5">satisfying_DES2</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V6" title="AFF_3:attr.6">satisfying_DES2_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V5" title="AFF_3:attr.5">satisfying_DES2</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#D5" target="_self" title="AFF_3:def.5">AFF_3:def 5</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; <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a9</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="b4">a9</font>,<font color="Olive" title="b8">c9</font> holds <br/><font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font></span><br/><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; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a9</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="b2">a9</font>,<font color="Olive" title="b6">c9</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font></span><br/><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; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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="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="c6">a9</font>,<font color="Maroon" title="c10">c9</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> )</span><br/>





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:23"/><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="E3:23"/><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="E4:23"/><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="E5:23"/><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="E6:23"/><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="E7:23"/><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="E8:23"/><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="E9:23"/><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="E10:23"/><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="E11:23"/><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="E12:23"/><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="E13:23"/><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="E14:23"/><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="E15:23"/><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="E16:23"/><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="E17:23"/><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="E18:23"/><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="E19:23"/><span class="lab"><font color="Green" title="E27">A19</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="c6">a9</font>
 <span class="kw">and </span><br/><a NAME="E20:23"/><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="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="E21:23"/><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="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="E22:23"/><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="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> )
 <span class="kw">and </span><br/><a NAME="E23:23"/><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="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"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span><br/>

<a NAME="E24:23"/><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="#E14:23"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:23"><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">P9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>;<br/>
<a NAME="E25:23"/><span class="lab"><font color="Green" title="E33">A25</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="c8">b9</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:23"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E26:23"/>
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:23"><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="E27:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A26</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</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#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>;<br/>
<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/>
<span class="kw">set </span><font color="Maroon" title="c16">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">T</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>;<br/>
<a NAME="E28:23"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:23"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E29:23"/><span class="lab"><font color="Green" title="E36">A28</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="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E30:23"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><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="E31:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A29</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E32:23"/>
<font color="Maroon" title="c8">b9</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="#E17:23"><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="E33:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A30</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</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#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E34:23"/><span class="lab"><font color="Green" title="E39">A31</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="#E16:23"><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="E35:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A32</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="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E36:23"/><span class="lab"><font color="Green" title="E41">A33</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="#E17:23"><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="E37:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A34</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#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E39:23"/><span class="lab"><font color="Green" title="E43">A35</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">M</font>
 <span class="kw">and </span><br/><a NAME="E40:23"/><span class="lab"><font color="Green" title="E44">A36</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> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c18">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E41:23"/><span class="lab"><font color="Green" title="E45">A37</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="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:23"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E34:23"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E36:23"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>;<br/>
<a NAME="E42:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A38</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> <font color="Maroon" title="c18">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:23"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E43:23"/><span class="lab"><font color="Green" title="E47">A39</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="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#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E44:23"/><span class="lab"><font color="Green" title="E48">A40</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">A9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>;<br/>
<a NAME="E45:23"/><span class="lab"><font color="Green" title="E49">A41</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="c7">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:23"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E46:23"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><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="E47:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A42</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E48:23"/><span class="lab"><font color="Green" title="E51">A43</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="c8">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E49:23"/><span class="lab"><font color="Green" title="E52">A44</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E50:23"/><span class="lab"><font color="Green" title="E53">A45</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="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E51:23"/><span class="lab"><font color="Green" title="E54">A46</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#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E52:23"/><span class="lab"><font color="Green" title="E55">A47</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="c8">b9</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E53:23"/><span class="lab"><font color="Green" title="E56">A48</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E54:23"/><span class="lab"><font color="Green" title="E57">A49</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="c7">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E55:23"/><span class="lab"><font color="Green" title="E58">A50</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E56:23"/><span class="lab"><font color="Green" title="E59">A51</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="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E57:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A52</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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E58:23"/><span class="lab"><font color="Green" title="E61">A53</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="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1"/>
<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="E2:23_1"/><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>  <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="#E56:23"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E41:23"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E19:23"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E35:23"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E51:23"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E50:23"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E57:23"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</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="E59:23"/><span class="lab"><font color="Green" title="E62">A54</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</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="E2:23_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E43:23"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="txt" href="#E48:23"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="txt" href="#E58:23"><span class="lab"><font color="Green" title="E61">A53</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="E3:23_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E24:23"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</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="E60:23"/><span class="lab"><font color="Green" title="E63">A55</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="23_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_3"/><span class="lab"><font color="Green" title="E64">A56</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_3_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</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="E2:23_3_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E19:23"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E54:23"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E52:23"><span class="lab"><font color="Green" title="E55">A47</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="E3:23_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</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>
<span class="kw">assume </span><a NAME="E2:23_3"/><span class="lab"><font color="Green" title="E65">A57</font></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>
 ; <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="E3:23_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</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="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E49:23"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E28:23"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E59:23"><span class="lab"><font color="Green" title="E62">A54</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="E4:23_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E45:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E53:23"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E2:23_3"><span class="lab"><font color="Green" title="E65">A57</font></span></a>, <a class="txt" href="#E1:23_3"><span class="lab"><font color="Green" title="E64">A56</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</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="E61:23"/><span class="lab"><font color="Green" title="E64">A58</font></span>: 
<font color="Maroon" title="c18">M</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="#E40:23"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E65">A59</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:23_4"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c18">M</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="c11">p</font>,<font color="Maroon" title="c12">q</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E3:23_4"/><span class="lab"><font color="Green" title="E66">A60</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E4:23_4"/><span class="lab"><font color="Green" title="E67">A61</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E61:23"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E5:23_4"/><span class="lab"><font color="Green" title="E68">A62</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="c20">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_4_1"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_4_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="c8">b9</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E49:23"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E53:23"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E3:23_4"><span class="lab"><font color="Green" title="E66">A60</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E3:23_4_1"/><span class="kw">then </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="c8">b9</font>,<font color="Maroon" title="c5">a</font> or <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="c8">b9</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E54:23"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E45:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E4:23_4_1"/><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="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E60:23"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:23"><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="E6:23_4"/>
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_4_2"/><span class="lab"><font color="Green" title="E69">A63</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_4_2"/>
<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="#E14:23"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:23"><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="E3:23_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_4_2"><span class="lab"><font color="Green" title="E69">A63</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></span>;<br/>
<a NAME="E4:23_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<a NAME="E5:23_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">M</font> or <font color="Maroon" title="c10">c9</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="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E49:23"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E48:23"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="txt" href="#E3:23_4"><span class="lab"><font color="Green" title="E66">A60</font></span></a>, <a class="txt" href="#E4:23_4"><span class="lab"><font color="Green" title="E67">A61</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E6:23_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> or <font color="Maroon" title="c18">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E24:23"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E53:23"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E61:23"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E7:23_4_2"/><span class="kw">then </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="c8">b9</font>,<font color="Maroon" title="c5">a</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:23"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E40:23"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E8:23_4_2"/><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> or  <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>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E50:23"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E52:23"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E60:23"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:23_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E36">A28</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#D2" title="AFF_1:def.2">AFF_1:def 2</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><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E8:23_4"/><span class="lab"><font color="Green" title="E69">A64</font></span>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E9:23_4"/><span class="lab"><font color="Green" title="E70">A65</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="c20">x</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></span>;<br/><a NAME="E10:23_4"/><span class="lab"><font color="Green" title="E71">A66</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="c21">y</font>,<font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23_4"><span class="lab"><font color="Green" title="E70">A65</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E11:23_4"/><span class="lab"><font color="Green" title="E72">A67</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="c21">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_3"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E73">A68</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:23_4_3_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:23_4_3_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="c8">b9</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E49:23"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E53:23"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E3:23_4"><span class="lab"><font color="Green" title="E66">A60</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E3:23_4_3_1"/><span class="kw">then </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="c8">b9</font>,<font color="Maroon" title="c5">a</font> or <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="c8">b9</font>,<font color="Maroon" title="c6">a9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E54:23"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E45:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E4:23_4_3_1"/><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="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E60:23"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:23_4_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:23"><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>
<span class="kw">assume </span><a NAME="E2:23_4_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c21">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:23_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A69</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="c21">y</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="E4:23_4_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E5:23_4_3"/><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="c21">y</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_4"><span class="lab"><font color="Green" title="E71">A66</font></span></a>, <a class="txt" href="#E3:23_4_3"><span class="lab"><font color="Green" title="E74">A69</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<a NAME="E6:23_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E24:23"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E8:23_4"><span class="lab"><font color="Green" title="E69">A64</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#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E7:23_4_3"/><span class="kw">then </span>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E45:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E61:23"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="txt" href="#E4:23_4"><span class="lab"><font color="Green" title="E67">A61</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E8:23_4_3"/><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>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:23"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E54:23"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E42:23"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:23_4_3"><span class="lab"><font color="Green" title="E73">A68</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:23_4_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E56:23"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</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="E12:23_4"/>
(  <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="c20">x</font> &amp; <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c20">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E50:23"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E49:23"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E48:23"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E40:23"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="txt" href="#E3:23_4"><span class="lab"><font color="Green" title="E66">A60</font></span></a>, <a class="txt" href="#E4:23_4"><span class="lab"><font color="Green" title="E67">A61</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E13:23_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</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="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E20:23"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E21:23"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E8:23_4"><span class="lab"><font color="Green" title="E69">A64</font></span></a>, <a class="txt" href="#E10:23_4"><span class="lab"><font color="Green" title="E71">A66</font></span></a>, <a class="txt" href="#E11:23_4"><span class="lab"><font color="Green" title="E72">A67</font></span></a>, <a class="txt" href="#E5:23_4"><span class="lab"><font color="Green" title="E68">A62</font></span></a>, <a class="ref" href="aff_3.html#D6" target="_self" title="AFF_3:def.6">Def6</a></span>;<br/><a NAME="E14:23_4"/><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="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:23"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E36:23"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E15:23_4"/><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="c21">y</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="E16:23_4"/><span class="kw">then </span>
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><a NAME="E17:23_4"/><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> or <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E35:23"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E56:23"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="txt" href="#E8:23_4"><span class="lab"><font color="Green" title="E69">A64</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E18:23_4"/><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="c4">C</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c20">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23_4"><span class="lab"><font color="Green" title="E70">A65</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#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E19:23_4"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E23">A15</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#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><a NAME="E20:23_4"/><span class="kw">then </span>
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">M</font> or <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="c7">b</font>,<font color="Maroon" title="c9">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E48:23"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="txt" href="#E28:23"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E3:23_4"><span class="lab"><font color="Green" title="E66">A60</font></span></a>, <a class="txt" href="#E4:23_4"><span class="lab"><font color="Green" title="E67">A61</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E21:23_4"/><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="c11">p</font>,<font color="Maroon" title="c12">q</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c7">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E51:23"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E56:23"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E43:23"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E42:23"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E22:23_4"/><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="c11">p</font>,<font color="Maroon" title="c12">q</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E58:23"><span class="lab"><font color="Green" title="E61">A53</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="E23:23_4"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E24:23"><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>
<div><span class="lab"><font color="Green" title="E66">A70</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:23_5"/>
not <font color="Maroon" title="c18">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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="c11">p</font>,<font color="Maroon" title="c12">q</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E3:23_5"/><span class="lab"><font color="Green" title="E67">A71</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">M</font>
 <span class="kw">and </span><br/><a NAME="E4:23_5"/><span class="lab"><font color="Green" title="E68">A72</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E61:23"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E5:23_5"/><span class="lab"><font color="Green" title="E69">A73</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="c20">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_5_1"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_5_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="c7">b</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E45:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E4:23_5"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E3:23_5_1"/><span class="kw">then </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="c8">b9</font>,<font color="Maroon" title="c6">a9</font> or <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E53:23"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E52:23"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E4:23_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</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="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E60:23"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:23"><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="E6:23_5"/>
not <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5_2"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E70">A74</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:23_5_2_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:23_5_2_1"/><span class="kw">then </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="c8">b9</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E53:23"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E52:23"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E3:23_5_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</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="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E60:23"><span class="lab"><font color="Green" title="E63">A55</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="E4:23_5_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:23"><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>
<span class="kw">assume </span><a NAME="E2:23_5_2"/><span class="lab"><font color="Green" title="E71">A75</font></span>: 
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:23_5_2"/>
<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="#E14:23"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:23"><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="E4:23_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c20">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_5_2"><span class="lab"><font color="Green" title="E71">A75</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></span>;<br/>
<a NAME="E5:23_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<a NAME="E6:23_5_2"/><span class="kw">then </span>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E4:23_5"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E7:23_5_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="c11">p</font> or <font color="Maroon" title="c18">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E24:23"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E43:23"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E45:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E61:23"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="txt" href="#E3:23_5"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E8:23_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:23"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E54:23"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E42:23"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:23_5_2"><span class="lab"><font color="Green" title="E70">A74</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:23_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E35:23"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E51:23"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E56:23"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</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><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E8:23_5"/><span class="lab"><font color="Green" title="E70">A76</font></span>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E9:23_5"/><span class="lab"><font color="Green" title="E71">A77</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="c20">x</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></span>;<br/><a NAME="E10:23_5"/><span class="lab"><font color="Green" title="E72">A78</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="c21">y</font>,<font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23_5"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E11:23_5"/><span class="lab"><font color="Green" title="E73">A79</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="c21">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5_3"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E74">A80</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:23_5_3_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:23_5_3_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="c7">b</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E47:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="txt" href="#E55:23"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E45:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E4:23_5"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E3:23_5_3_1"/><span class="kw">then </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="c8">b9</font>,<font color="Maroon" title="c6">a9</font> or <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E53:23"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E52:23"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E4:23_5_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</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="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E60:23"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:23_5_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:23"><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>
<span class="kw">assume </span><a NAME="E2:23_5_3"/>
 <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="c21">y</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="E3:23_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A81</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="c21">y</font>,<font color="Maroon" title="c6">a9</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="E4:23_5_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E5:23_5_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c20">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_5"><span class="lab"><font color="Green" title="E72">A78</font></span></a>, <a class="txt" href="#E3:23_5_3"><span class="lab"><font color="Green" title="E75">A81</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<a NAME="E6:23_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E24:23"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E8:23_5"><span class="lab"><font color="Green" title="E70">A76</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#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E7:23_5_3"/><span class="kw">then </span>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> or  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:23"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E25:23"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E61:23"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="txt" href="#E3:23_5"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E8:23_5_3"/><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>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:23"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E52:23"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E40:23"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="txt" href="#E1:23_5_3"><span class="lab"><font color="Green" title="E74">A80</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:23_5_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</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="E12:23_5"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c20">x</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c20">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E51:23"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E56:23"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E43:23"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E42:23"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E3:23_5"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="txt" href="#E4:23_5"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E13:23_5"/><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="c6">a9</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E20:23"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E21:23"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E8:23_5"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E10:23_5"><span class="lab"><font color="Green" title="E72">A78</font></span></a>, <a class="txt" href="#E11:23_5"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="txt" href="#E5:23_5"><span class="lab"><font color="Green" title="E69">A73</font></span></a>, <a class="ref" href="aff_3.html#D6" target="_self" title="AFF_3:def.6">Def6</a></span>;<br/><a NAME="E14:23_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c21">y</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E23:23"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E34:23"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E15:23_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c10">c9</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="E16:23_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c21">y</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="E17:23_5"/><span class="kw">then </span>
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><a NAME="E18:23_5"/><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> or <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E37:23"><span class="lab"><font color="Green" title="E42">A34</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E8:23_5"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E19:23_5"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c20">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23_5"><span class="lab"><font color="Green" title="E71">A77</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#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E20:23_5"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E23">A15</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#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><a NAME="E21:23_5"/><span class="kw">then </span>
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">M</font> or <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="c7">b</font>,<font color="Maroon" title="c9">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E33:23"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E48:23"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="txt" href="#E28:23"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E3:23_5"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="txt" href="#E4:23_5"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E22:23_5"/><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="c11">p</font>,<font color="Maroon" title="c12">q</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c7">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E51:23"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E56:23"><span class="lab"><font color="Green" title="E59">A51</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E43:23"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E42:23"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E23:23_5"/><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="c11">p</font>,<font color="Maroon" title="c12">q</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E58:23"><span class="lab"><font color="Green" title="E61">A53</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="E24:23_5"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E24:23"><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="E64:23"/>
(  not <font color="Maroon" title="c18">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> or  not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c18">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_6"/>
( <font color="Maroon" title="c18">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c18">M</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="E2:23_6"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:23"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E59:23"><span class="lab"><font color="Green" title="E62">A54</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>
<span class="kw">hence </span><a NAME="E65:23"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:23"><span class="lab"><font color="Green" title="E66">A70</font></span></a>, <a class="txt" href="#E62:23"><span class="lab"><font color="Green" title="E65">A59</font></span></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>
