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<span class="kw">let </span><font color="Maroon" title="c1">AFP</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"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">p9</font>, <font color="Olive" title="b4">q9</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">x</font>, <font color="Olive" title="b8">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><br/> for <font color="Olive" title="b9">K</font>, <font color="Olive" title="b10">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b9">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">K</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">K</font> &amp; <font color="Olive" title="b3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">K</font> &amp; <font color="Olive" title="b4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">K</font> &amp; <font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>  &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">p9</font>,<font color="Olive" title="b7">x</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">p9</font>,<font color="Olive" title="b8">y</font> &amp; <font color="Olive" title="b2">q</font>,<font color="Olive" title="b5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">q9</font>,<font color="Olive" title="b7">x</font> holds <br/><font color="Olive" title="b2">q</font>,<font color="Olive" title="b6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">q9</font>,<font color="Olive" title="b8">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">p9</font>, <font color="Maroon" title="c5">q9</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</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">K</font>, <font color="Olive" title="b2">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">M</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c4">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c5">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>  &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c8">x</font> holds <br/><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">K</font>, <font color="Maroon" title="c11">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</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="c10">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c4">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c5">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>  &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c8">x</font> implies <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font> )</span><br/>





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c10">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">and </span><br/><a NAME="E3:27"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">and </span><br/><a NAME="E4:27"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">and </span><br/><a NAME="E5:27"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c5">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">and </span><br/><a NAME="E6:27"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a> 
 <span class="kw">and </span><br/><a NAME="E7:27"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E8:27"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E9:27"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E10:27"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E11:27"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E12:27"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E13:27"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E14:27"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">and </span><br/><a NAME="E15:27"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c8">x</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="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/>

<a NAME="E16:27"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c10">K</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="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<a NAME="E17:27"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c11">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="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c12">D9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c8">x</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">C9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>;<br/><span class="kw">set </span><font color="Maroon" title="c14">B99</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c9">y</font>;<br/><span class="kw">set </span><font color="Maroon" title="c15">A9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c8">x</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="c3">q</font>,<font color="Maroon" title="c6">a</font>;<br/><span class="kw">set </span><font color="Maroon" title="c17">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c18">B9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>;<br/><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c3">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="c3">q</font>,<font color="Maroon" title="c6">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="E2:27_1"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">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="E3:27_1"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c8">x</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="E4:27_1"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
<font color="Maroon" title="c5">q9</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">q9</font>,<font color="Maroon" title="c8">x</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="E5:27_1"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E6:27_1"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><span class="kw">assume </span><a NAME="E7:27_1"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
<font color="Maroon" title="c10">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">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="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/><a NAME="E8:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<font color="Maroon" title="c5">q9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E9:27_1"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E10:27_1"/><span class="lab"><font color="Green" title="E54">A27</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="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E11:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A28</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</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="E12:27_1"/><span class="lab"><font color="Green" title="E56">A29</font></span>: 
not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E13:27_1"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E14:27_1"/><span class="lab"><font color="Green" title="E58">A31</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><div><span class="lab"><font color="Green" title="E59">A32</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E60">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_1_1"/><span class="lab"><font color="Green" title="E61">A34</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1_1_1"/><span class="lab"><font color="Green" title="E62">A35</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</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="E3:27_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E2:27_1_1_1"><span class="lab"><font color="Green" title="E62">A35</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:27_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E5:27_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E12:27_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E6:27_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E1:27_1_1_1"><span class="lab"><font color="Green" title="E61">A34</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E7:27_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E8:27_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">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/><span class="kw">hence </span><a NAME="E9:27_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:27_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</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="E61">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_1_2"/><span class="lab"><font color="Green" title="E62">A37</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1_1_2"/><span class="lab"><font color="Green" title="E63">A38</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</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="E3:27_1_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E1:27_1_1_2"><span class="lab"><font color="Green" title="E62">A37</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:27_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E5:27_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E13:27_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E6:27_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E7:27_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">q9</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="E8:27_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1_2"><span class="lab"><font color="Green" title="E63">A38</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><span class="kw">hence </span><a NAME="E9:27_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:27_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</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="E3:27_1_1"/><span class="lab"><font color="Green" title="E62">A39</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  holds <br/>(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">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="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/><div><span class="lab"><font color="Green" title="E63">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_1_3"/><span class="lab"><font color="Green" title="E64">A41</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1_1_3"/><span class="lab"><font color="Green" title="E65">A42</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</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="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/><div><span class="lab"><font color="Green" title="E66">A43</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_1_1_3_1"/><span class="lab"><font color="Green" title="E67">A44</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</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="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/><a NAME="E2:27_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c9">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E10:27_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E1:27_1_1_3"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="txt" href="#E2:27_1_1_3"><span class="lab"><font color="Green" title="E65">A42</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E3:27_1_1_3_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:27_1_1_3_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E5:27_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E10:27"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E12:27_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E1:27_1_1_3"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="txt" href="#E2:27_1_1_3"><span class="lab"><font color="Green" title="E65">A42</font></span></a>, <a class="txt" href="#E1:27_1_1_3_1"><span class="lab"><font color="Green" title="E67">A44</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="E6:27_1_1_3_1"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>; <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><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_1_1_3_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="E2:27_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E1:27_1_1_3"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E3:27_1_1_3_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:27_1_1_3_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">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/><span class="kw">hence </span><a NAME="E5:27_1_1_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E9:27_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</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="E5:27_1_1_3"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E14:27_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="txt" href="#E3:27_1_1_3"><span class="lab"><font color="Green" title="E66">A43</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><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_1_4"/><span class="lab"><font color="Green" title="E64">A45</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1_1_4"/><span class="lab"><font color="Green" title="E65">A46</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</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="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/><a NAME="E3:27_1_1_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E1:27_1_1_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:27_1_1_4"/><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="c9">y</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E5:27_1_1_4"/><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="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E10:27"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E13:27_1"><span class="lab"><font color="Green" title="E57">A30</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="E6:27_1_1_4"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1_4"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</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="E6:27_1_1"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E61">A36</font></span></a>, <a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="txt" href="#E4:27_1_1"><span class="lab"><font color="Green" title="E63">A40</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><span class="kw">end;</span></div><a NAME="E16:27_1"/><span class="lab"><font color="Green" title="E60">A47</font></span>: 
<font color="Maroon" title="c9">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="c4">p9</font>,<font color="Maroon" title="c9">y</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="E17:27_1"/><span class="lab"><font color="Green" title="E61">A48</font></span>: 
<font color="Maroon" title="c4">p9</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="c4">p9</font>,<font color="Maroon" title="c9">y</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="E18:27_1"/><span class="lab"><font color="Green" title="E62">A49</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="c2">p</font>,<font color="Maroon" title="c7">b</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="E19:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A50</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E20:27_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E21:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A51</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c6">a</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="c5">q9</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>;<br/><a NAME="E22:27_1"/><span class="lab"><font color="Green" title="E65">A52</font></span>: 
<font color="Maroon" title="c2">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="c2">p</font>,<font color="Maroon" title="c7">b</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="E23:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A53</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E24:27_1"/><span class="lab"><font color="Green" title="E67">A54</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E10:27"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E25:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A55</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</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="c4">p9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E10:27_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>;<br/><a NAME="E26:27_1"/><span class="lab"><font color="Green" title="E69">A56</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c9">y</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="#E24:27_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:27_1_2"/><span class="lab"><font color="Green" title="E70">A57</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E21:27_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><span class="kw">assume </span><a NAME="E2:27_1_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</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">b</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="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">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="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E4:27_1_2"/><span class="lab"><font color="Green" title="E71">A58</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">d</font>
 <span class="kw">and </span><br/><a NAME="E5:27_1_2"/><span class="lab"><font color="Green" title="E72">A59</font></span>: 
<font color="Maroon" title="c20">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E6:27_1_2"/><span class="lab"><font color="Green" title="E73">A60</font></span>: 
<font color="Maroon" title="c20">d</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="ref" href="translac.html#T2" title="TRANSLAC:th.2">TRANSLAC:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E8:27_1_2"/><span class="lab"><font color="Green" title="E74">A61</font></span>: 
<font color="Maroon" title="c20">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 <span class="kw">and </span><br/><a NAME="E9:27_1_2"/><span class="lab"><font color="Green" title="E75">A62</font></span>: 
<font color="Maroon" title="c10">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/><a NAME="E10:27_1_2"/><span class="lab"><font color="Green" title="E76">A63</font></span>: 
<font color="Maroon" title="c20">d</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="c2">p</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2"><span class="lab"><font color="Green" title="E71">A58</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="E11:27_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A64</font></span>: 
<font color="Maroon" title="c21">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E11:27_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E18:27_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E23:27_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="txt" href="#E6:27_1_2"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E74">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="E12:27_1_2"/><span class="lab"><font color="Green" title="E78">A65</font></span>: 
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c9">y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:27_1_2_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c9">y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N</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="E2:27_1_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:27_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:27_1_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27_1_2"><span class="lab"><font color="Green" title="E75">A62</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">hence </span><a NAME="E4:27_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E22:27_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E19:27_1"><span class="lab"><font color="Green" title="E63">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><a NAME="E13:27_1_2"/>
<font color="Maroon" title="c21">N</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="#E9:27_1_2"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E15:27_1_2"/><span class="lab"><font color="Green" title="E79">A66</font></span>: 
<font color="Maroon" title="c22">z</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="c4">p9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">and </span><br/><a NAME="E16:27_1_2"/><span class="lab"><font color="Green" title="E80">A67</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:27_1"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="txt" href="#E12:27_1_2"><span class="lab"><font color="Green" title="E78">A65</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E17:27_1_2"/><span class="lab"><font color="Green" title="E81">A68</font></span>: 
<font color="Maroon" title="c20">d</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">z</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:27_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E16:27_1"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="txt" href="#E25:27_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E10:27_1_2"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E15:27_1_2"><span class="lab"><font color="Green" title="E79">A66</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E18:27_1_2"/><span class="lab"><font color="Green" title="E82">A69</font></span>: 
<font color="Maroon" title="c21">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:27_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E22:27_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E19:27_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E5:27_1_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E10:27_1_2"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E74">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="E19:27_1_2"/><span class="lab"><font color="Green" title="E83">A70</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E9:27_1_2"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E20:27_1_2"/><span class="lab"><font color="Green" title="E84">A71</font></span>: 
<font color="Maroon" title="c10">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E11:27_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E22:27_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E19:27_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E5:27_1_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E10:27_1_2"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E74">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="E21:27_1_2"/><span class="lab"><font color="Green" title="E85">A72</font></span>: 
<font color="Maroon" title="c21">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E9:27_1_2"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E22:27_1_2"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c20">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c22">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:27_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E17:27_1"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E25:27_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E10:27_1_2"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E15:27_1_2"><span class="lab"><font color="Green" title="E79">A66</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E23:27_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A73</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c20">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c22">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="txt" href="#E9:27_1_2"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E16:27_1_2"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E20:27_1_2"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="txt" href="homothet.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>;<br/><a NAME="E24:27_1_2"/>
<font color="Maroon" title="c11">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E11:27_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E18:27_1"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E23:27_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="txt" href="#E6:27_1_2"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E10:27_1_2"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E74">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="E25:27_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">d</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">z</font>,<font color="Maroon" title="c5">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="txt" href="#E16:27_1_2"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E23:27_1_2"><span class="lab"><font color="Green" title="E86">A73</font></span></a>, <a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="txt" href="#E19:27_1_2"><span class="lab"><font color="Green" title="E83">A70</font></span></a>, <a class="txt" href="homothet.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E26:27_1_2"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E10:27"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="txt" href="#E9:27_1_2"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E16:27_1_2"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E17:27_1_2"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="txt" href="#E21:27_1_2"><span class="lab"><font color="Green" title="E85">A72</font></span></a>, <a class="txt" href="#E11:27_1_2"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="txt" href="#E18:27_1_2"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="homothet.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E28:27_1"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27_1"><span class="lab"><font color="Green" title="E59">A32</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E19:27"/>
<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E10:27"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</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/>


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