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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="conaffm.html#V1" title="CONAFFM:attr.1">satisfying_DES</a>  iff  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  )</span><br/>

<a NAME="E1:23"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="conaffm.html#V1" title="CONAFFM:attr.1">satisfying_DES</a>  implies  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  )
 
<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"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conaffm.html#V1" title="CONAFFM:attr.1">satisfying_DES</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">a1</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">M</font> &amp; <font color="Olive" title="b6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b7">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <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">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">N</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">M</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">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b7">c1</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b1</font>,<font color="Olive" title="b7">c1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c11">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a1</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">M</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <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">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">N</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">M</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">N</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c11">c1</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">c1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_1_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c5">o</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="E2:23_1_1"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E3:23_1_1"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1_1"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E5:23_1_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c5">o</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="E6:23_1_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E7:23_1_1"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E8:23_1_1"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c9">a1</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_1_1"/><span class="lab"><font color="Green" title="E25">A11</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">M</font>
 <span class="kw">and </span><br/><a NAME="E10:23_1_1"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c10">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E11:23_1_1"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E12:23_1_1"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E13:23_1_1"/><span class="lab"><font color="Green" title="E29">A15</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="E14:23_1_1"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E15:23_1_1"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<font color="Maroon" title="c4">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E16:23_1_1"/><span class="lab"><font color="Green" title="E32">A18</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">M</font>
 <span class="kw">and </span><br/><a NAME="E17:23_1_1"/><span class="lab"><font color="Green" title="E33">A19</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">N</font>
 <span class="kw">and </span><br/><a NAME="E18:23_1_1"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font>
 <span class="kw">and </span><br/><a NAME="E19:23_1_1"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c11">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">c1</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">o9</font> = <font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c13">a9</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c14">a19</font> = <font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c15">b9</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c16">b19</font> = <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c17">c9</font> = <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c18">c19</font> = <font color="Maroon" title="c11">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T35" title="ANALMETR:th.35">ANALMETR:35</a></span>;<br/><a NAME="E21:23_1_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E10:23_1_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/><a NAME="E22:23_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b1</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="E23:23_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A23</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">o9</font>,<font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c16">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><a NAME="E24:23_1_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/><a NAME="E25:23_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A24</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a1</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="E26:23_1_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:23_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E12:23_1_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E15:23_1_1"><span class="lab"><font color="Green" title="E31">A17</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/><a NAME="E27:23_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A25</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c1</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="E28:23_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A26</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">o9</font>,<font color="Maroon" title="c17">c9</font>,<font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><span class="kw">assume </span><a NAME="E29:23_1_1"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
 not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E30:23_1_1"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
(  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c16">b19</font>,<font color="Maroon" title="c13">a9</font> &amp;  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c13">a9</font>,<font color="Maroon" title="c14">a19</font>,<font color="Maroon" title="c17">c9</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1_1_1"/><span class="lab"><font color="Green" title="E43">A29</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="c9">a1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_1_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E3:23_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A33</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E4:23_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:23_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E15:23_1_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E17:23_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E3:23_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  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_1_1_1_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E3:23_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:23_1_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E27:23_1_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:23_1_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_1_1_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E3:23_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E4:23_1_1_1_1_2_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_1_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E16:23_1_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E3:23_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  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_1_1_1_1_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E3:23_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23_1_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E22:23_1_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:23_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23_1_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E2:23_1_1_1_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="aff_1.html#T2" title="AFF_1:th.2">AFF_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:23_1_1_1"/><span class="lab"><font color="Green" title="E44">A36</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="c10">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A37</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_1_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E3:23_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A38</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E4:23_1_1_1_2_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E16:23_1_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E3:23_1_1_1_2_1"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  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_1_1_1_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:23_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23_1_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:23_1_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E1:23_1_1_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E1:23_1_1_1_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:23_1_1_1"/>
(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c16">b19</font>,<font color="Maroon" title="c13">a9</font> or  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c13">a9</font>,<font color="Maroon" title="c14">a19</font>,<font color="Maroon" title="c17">c9</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:23_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="c7">b</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c6">a</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c8">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/>
<a NAME="E5:23_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> or <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E10:23_1_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E1:23_1_1_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E2:23_1_1_1"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E6:23_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A39</font></span>: 
( <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font> or <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E7:23_1_1_1"/><span class="lab"><font color="Green" title="E46">A40</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:23_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E15:23_1_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E8:23_1_1_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:23_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E16:23_1_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E17:23_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E6:23_1_1_1"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E7:23_1_1_1"><span class="lab"><font color="Green" title="E46">A40</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E31:23_1_1"/><span class="lab"><font color="Green" title="E43">A41</font></span>: 
<font color="Maroon" title="c13">a9</font>,<font color="Maroon" title="c17">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a19</font>,<font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:23_1_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/><a NAME="E32:23_1_1"/><span class="lab"><font color="Green" title="E44">A42</font></span>: 
<font color="Maroon" title="c13">a9</font>,<font color="Maroon" title="c15">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a19</font>,<font color="Maroon" title="c16">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23_1_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/><a NAME="E33:23_1_1"/><span class="lab"><font color="Green" title="E45">A43</font></span>: 
( <font color="Maroon" title="c12">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a19</font> &amp; <font color="Maroon" title="c12">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b19</font> &amp; <font color="Maroon" title="c12">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">c19</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E46">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:23_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A45</font></span>: 
<font color="Maroon" title="c12">o9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:23_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A46</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1_1_2_1_1"/><span class="lab"><font color="Green" title="E49">A47</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_2_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_2_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="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E3:23_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E4:23_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A48</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E5:23_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:23_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E15:23_1_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E17:23_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E4:23_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A48</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  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_1_1_2_1_1"/><span class="lab"><font color="Green" title="E50">A49</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:23_1_1_2_1_1"/><span class="lab"><font color="Green" title="E51">A50</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:23_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E12:23_1_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E15:23_1_1"><span class="lab"><font color="Green" title="E31">A17</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/>
<a NAME="E4:23_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:23_1_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E1:23_1_1_2_1"><span class="lab"><font color="Green" title="E47">A45</font></span></a>, <a class="txt" href="#E2:23_1_1_2_1_1"><span class="lab"><font color="Green" title="E50">A49</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E5:23_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E6:23_1_1_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</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="E7:23_1_1_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E8:23_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E9:23_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E3:23_1_1_2_1_1"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E10:23_1_1_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c1</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="E11:23_1_1_2_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="c5">o</font>,<font color="Maroon" title="c11">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E12:23_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A51</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c1</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="E13:23_1_1_2_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:23_1_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:23_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1_2_1_1"><span class="lab"><font color="Green" title="E50">A49</font></span></a>, <a class="txt" href="#E1:23_1_1_2_1_1"><span class="lab"><font color="Green" title="E49">A47</font></span></a>, <a class="txt" href="#E12:23_1_1_2_1_1"><span class="lab"><font color="Green" title="E52">A51</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:23_1_1_2_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1_1_2_1_2"/><span class="lab"><font color="Green" title="E49">A52</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="c6">a</font>,<font color="Maroon" title="c5">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_2_1_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_2_1_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E3:23_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E4:23_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A53</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E5:23_1_1_2_1_2_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E16:23_1_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E4:23_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  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_1_1_2_1_2"/><span class="lab"><font color="Green" title="E50">A54</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:23_1_1_2_1_2"/><span class="lab"><font color="Green" title="E51">A55</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E10:23_1_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/>
<a NAME="E4:23_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23_1_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E1:23_1_1_2_1"><span class="lab"><font color="Green" title="E47">A45</font></span></a>, <a class="txt" href="#E2:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E50">A54</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E5:23_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E6:23_1_1_2_1_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="c6">a</font>,<font color="Maroon" title="c5">o</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="E7:23_1_1_2_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E8:23_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E9:23_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E3:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A55</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E10:23_1_1_2_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b1</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="E11:23_1_1_2_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="c5">o</font>,<font color="Maroon" title="c10">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E12:23_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A56</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b1</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="E13:23_1_1_2_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:23_1_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:23_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E50">A54</font></span></a>, <a class="txt" href="#E1:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E49">A52</font></span></a>, <a class="txt" href="#E12:23_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A56</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:23_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23_1_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E2:23_1_1_2_1"><span class="lab"><font color="Green" title="E48">A46</font></span></a>, <a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E47">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:23_1_1_2_2"/><span class="lab"><font color="Green" title="E48">A58</font></span>: 
 not  <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="c5">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_2_2_1"/>
 <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="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_2_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E3:23_1_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E4:23_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A59</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E5:23_1_1_2_2_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E14:23_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23_1_1_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E16:23_1_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E4:23_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A59</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:23_1_1_2_2"/><span class="lab"><font color="Green" title="E49">A60</font></span>: 
<font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/><span class="kw">assume </span><a NAME="E3:23_1_1_2_2"/>
<font color="Maroon" title="c12">o9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:23_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23_1_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="txt" href="#E2:23_1_1_2_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><a NAME="E5:23_1_1_2_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="c7">b</font>,<font color="Maroon" title="c5">o</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="E6:23_1_1_2_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E7:23_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</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="E8:23_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/><a NAME="E9:23_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E2:23_1_1_2_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><a NAME="E10:23_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/><a NAME="E11:23_1_1_2_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a1</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="E12:23_1_1_2_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="c5">o</font>,<font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E13:23_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A61</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a1</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="E14:23_1_1_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:23_1_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E15:23_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="txt" href="#E1:23_1_1_2_2"><span class="lab"><font color="Green" title="E48">A58</font></span></a>, <a class="txt" href="#E13:23_1_1_2_2"><span class="lab"><font color="Green" title="E50">A61</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E48">A62</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:23_1_1_2_3"/><span class="lab"><font color="Green" title="E49">A63</font></span>: 
 not  <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">c</font>,<font color="Maroon" title="c5">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_2_3_1"/>
 <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">c</font>,<font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_1_1_2_3_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E3:23_1_1_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E4:23_1_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A64</font></span>: 
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E5:23_1_1_2_3_1"/>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E11:23_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E15:23_1_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23_1_1_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E17:23_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E4:23_1_1_2_3_1"><span class="lab"><font color="Green" title="E50">A64</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:23_1_1_2_3"/><span class="lab"><font color="Green" title="E50">A65</font></span>: 
<font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E7:23_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E13:23_1_1"><span class="lab"><font color="Green" title="E29">A15</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/><span class="kw">assume </span><a NAME="E3:23_1_1_2_3"/>
<font color="Maroon" title="c12">o9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:23_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:23_1_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="txt" href="#E2:23_1_1_2_3"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><a NAME="E5:23_1_1_2_3"/><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">c</font>,<font color="Maroon" title="c5">o</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="E6:23_1_1_2_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E7:23_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</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="E8:23_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/><a NAME="E9:23_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E2:23_1_1_2_3"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><a NAME="E10:23_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/><a NAME="E11:23_1_1_2_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a1</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="E12:23_1_1_2_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E13:23_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A66</font></span>: 
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a1</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="E14:23_1_1_2_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:23_1_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E15:23_1_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="txt" href="#E1:23_1_1_2_3"><span class="lab"><font color="Green" title="E49">A63</font></span></a>, <a class="txt" href="#E13:23_1_1_2_3"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E4:23_1_1_2"/>
(  not <font color="Maroon" title="c12">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a19</font> or  not <font color="Maroon" title="c12">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b19</font> or  not <font color="Maroon" title="c12">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">c19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E5:23_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="txt" href="#E2:23_1_1_2"><span class="lab"><font color="Green" title="E47">A57</font></span></a>, <a class="txt" href="#E3:23_1_1_2"><span class="lab"><font color="Green" title="E48">A62</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E34:23_1_1"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">o9</font>,<font color="Maroon" title="c13">a9</font>,<font color="Maroon" title="c14">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:23_1_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><a NAME="E35:23_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c17">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">b19</font>,<font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E23:23_1_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E28:23_1_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E33:23_1_1"><span class="lab"><font color="Green" title="E45">A43</font></span></a>, <a class="txt" href="#E30:23_1_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E32:23_1_1"><span class="lab"><font color="Green" title="E44">A42</font></span></a>, <a class="txt" href="#E31:23_1_1"><span class="lab"><font color="Green" title="E43">A41</font></span></a>, <a class="ref" href="conaffm.html#D1" title="CONAFFM:def.1">CONAFFM:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E36:23_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23_1_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:23_1"/>
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:23"/>
(  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="conaffm.html#V1" title="CONAFFM:attr.1">satisfying_DES</a>  )
 
<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"/><span class="lab"><font color="Green" title="E16">A67</font></span>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="conaffm.html#V1" title="CONAFFM:attr.1">satisfying_DES</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">o9</font>, <font color="Maroon" title="c3">a9</font>, <font color="Maroon" title="c4">a19</font>, <font color="Maroon" title="c5">b9</font>, <font color="Maroon" title="c6">b19</font>, <font color="Maroon" title="c7">c9</font>, <font color="Maroon" title="c8">c19</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a9</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a19</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b9</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c19</font> &amp;  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c3">a9</font> &amp;  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">a19</font>,<font color="Maroon" title="c7">c9</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">o9</font>,<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">a19</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">o9</font>,<font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c6">b19</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">o9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">c19</font> &amp; <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c5">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a19</font>,<font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a19</font>,<font color="Maroon" title="c8">c19</font> implies <font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c8">c19</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_2_1"/><span class="lab"><font color="Green" title="E17">A68</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a9</font>
 <span class="kw">and </span><br/><a NAME="E2:23_2_1"/>
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a19</font>
 <span class="kw">and </span><br/><a NAME="E3:23_2_1"/><span class="lab"><font color="Green" title="E18">A69</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b9</font>
 <span class="kw">and </span><br/><a NAME="E4:23_2_1"/>
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b19</font>
 <span class="kw">and </span><br/><a NAME="E5:23_2_1"/><span class="lab"><font color="Green" title="E19">A70</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font>
 <span class="kw">and </span><br/><a NAME="E6:23_2_1"/>
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c19</font>
 <span class="kw">and </span><br/><a NAME="E7:23_2_1"/><span class="lab"><font color="Green" title="E20">A71</font></span>: 
 not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c3">a9</font>
 <span class="kw">and </span><br/><a NAME="E8:23_2_1"/><span class="lab"><font color="Green" title="E21">A72</font></span>: 
 not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">a19</font>,<font color="Maroon" title="c7">c9</font>
 <span class="kw">and </span><br/><a NAME="E9:23_2_1"/><span class="lab"><font color="Green" title="E22">A73</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">o9</font>,<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">a19</font>
 <span class="kw">and </span><br/><a NAME="E10:23_2_1"/><span class="lab"><font color="Green" title="E23">A74</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">o9</font>,<font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c6">b19</font>
 <span class="kw">and </span><br/><a NAME="E11:23_2_1"/><span class="lab"><font color="Green" title="E24">A75</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">o9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">c19</font>
 <span class="kw">and </span><br/><a NAME="E12:23_2_1"/><span class="lab"><font color="Green" title="E25">A76</font></span>: 
<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c5">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a19</font>,<font color="Maroon" title="c6">b19</font>
 <span class="kw">and </span><br/><a NAME="E13:23_2_1"/><span class="lab"><font color="Green" title="E26">A77</font></span>: 
<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a19</font>,<font color="Maroon" title="c8">c19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c8">c19</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">o</font> = <font color="Maroon" title="c2">o9</font>, <font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c3">a9</font>, <font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c4">a19</font>, <font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c5">b9</font>, <font color="Maroon" title="c13">b1</font> = <font color="Maroon" title="c6">b19</font>, <font color="Maroon" title="c14">c</font> = <font color="Maroon" title="c7">c9</font>, <font color="Maroon" title="c15">c1</font> = <font color="Maroon" title="c8">c19</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T35" title="ANALMETR:th.35">ANALMETR:35</a></span>;<br/><a NAME="E15:23_2_1"/><span class="lab"><font color="Green" title="E27">A78</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c14">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_2_1"><span class="lab"><font color="Green" title="E21">A72</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><a NAME="E16:23_2_1"/><span class="lab"><font color="Green" title="E28">A79</font></span>: 
<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c14">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:23_2_1"><span class="lab"><font color="Green" title="E26">A77</font></span></a>, <a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/><a NAME="E17:23_2_1"/><span class="lab"><font color="Green" title="E29">A80</font></span>: 
<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c13">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23_2_1"><span class="lab"><font color="Green" title="E25">A76</font></span></a>, <a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/><a NAME="E18:23_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c13">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_2_1"><span class="lab"><font color="Green" title="E23">A74</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E20:23_2_1"/><span class="lab"><font color="Green" title="E30">A81</font></span>: 
<font color="Maroon" title="c16">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E21:23_2_1"/><span class="lab"><font color="Green" title="E31">A82</font></span>: 
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M</font>
 <span class="kw">and </span><br/><a NAME="E22:23_2_1"/><span class="lab"><font color="Green" title="E32">A83</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M</font>
 <span class="kw">and </span><br/><a NAME="E23:23_2_1"/><span class="lab"><font color="Green" title="E33">A84</font></span>: 
<font color="Maroon" title="c13">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/><a NAME="E24:23_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23_2_1"><span class="lab"><font color="Green" title="E24">A75</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E26:23_2_1"/><span class="lab"><font color="Green" title="E34">A85</font></span>: 
<font color="Maroon" title="c17">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E27:23_2_1"/><span class="lab"><font color="Green" title="E35">A86</font></span>: 
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 <span class="kw">and </span><br/><a NAME="E28:23_2_1"/><span class="lab"><font color="Green" title="E36">A87</font></span>: 
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 <span class="kw">and </span><br/><a NAME="E29:23_2_1"/><span class="lab"><font color="Green" title="E37">A88</font></span>: 
<font color="Maroon" title="c15">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/><a NAME="E30:23_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23_2_1"><span class="lab"><font color="Green" title="E22">A73</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E32:23_2_1"/><span class="lab"><font color="Green" title="E38">A89</font></span>: 
<font color="Maroon" title="c18">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="E33:23_2_1"/><span class="lab"><font color="Green" title="E39">A90</font></span>: 
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">and </span><br/><a NAME="E34:23_2_1"/><span class="lab"><font color="Green" title="E40">A91</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">and </span><br/><a NAME="E35:23_2_1"/><span class="lab"><font color="Green" title="E41">A92</font></span>: 
<font color="Maroon" title="c11">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/><a NAME="E36:23_2_1"/><span class="lab"><font color="Green" title="E42">A93</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c13">b1</font>,<font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_2_1"><span class="lab"><font color="Green" title="E20">A71</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/><a NAME="E37:23_2_1"/>
( <font color="Maroon" title="c18">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">M</font> &amp; <font color="Maroon" title="c18">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">N</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_2_1_1"/>
(  not <font color="Maroon" title="c18">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">M</font> or  not <font color="Maroon" title="c18">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">N</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:23_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c13">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c10">a</font> or <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c14">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:23_2_1"><span class="lab"><font color="Green" title="E38">A89</font></span></a>, <a class="txt" href="#E34:23_2_1"><span class="lab"><font color="Green" title="E40">A91</font></span></a>, <a class="txt" href="#E35:23_2_1"><span class="lab"><font color="Green" title="E41">A92</font></span></a>, <a class="txt" href="#E22:23_2_1"><span class="lab"><font color="Green" title="E32">A83</font></span></a>, <a class="txt" href="#E23:23_2_1"><span class="lab"><font color="Green" title="E33">A84</font></span></a>, <a class="txt" href="#E28:23_2_1"><span class="lab"><font color="Green" title="E36">A87</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#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:23_2_1"><span class="lab"><font color="Green" title="E42">A93</font></span></a>, <a class="txt" href="#E15:23_2_1"><span class="lab"><font color="Green" title="E27">A78</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E38:23_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c14">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">b1</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_2"><span class="lab"><font color="Green" title="E16">A67</font></span></a>, <a class="txt" href="#E1:23_2_1"><span class="lab"><font color="Green" title="E17">A68</font></span></a>, <a class="txt" href="#E3:23_2_1"><span class="lab"><font color="Green" title="E18">A69</font></span></a>, <a class="txt" href="#E5:23_2_1"><span class="lab"><font color="Green" title="E19">A70</font></span></a>, <a class="txt" href="#E32:23_2_1"><span class="lab"><font color="Green" title="E38">A89</font></span></a>, <a class="txt" href="#E33:23_2_1"><span class="lab"><font color="Green" title="E39">A90</font></span></a>, <a class="txt" href="#E34:23_2_1"><span class="lab"><font color="Green" title="E40">A91</font></span></a>, <a class="txt" href="#E35:23_2_1"><span class="lab"><font color="Green" title="E41">A92</font></span></a>, <a class="txt" href="#E20:23_2_1"><span class="lab"><font color="Green" title="E30">A81</font></span></a>, <a class="txt" href="#E21:23_2_1"><span class="lab"><font color="Green" title="E31">A82</font></span></a>, <a class="txt" href="#E22:23_2_1"><span class="lab"><font color="Green" title="E32">A83</font></span></a>, <a class="txt" href="#E23:23_2_1"><span class="lab"><font color="Green" title="E33">A84</font></span></a>, <a class="txt" href="#E26:23_2_1"><span class="lab"><font color="Green" title="E34">A85</font></span></a>, <a class="txt" href="#E27:23_2_1"><span class="lab"><font color="Green" title="E35">A86</font></span></a>, <a class="txt" href="#E28:23_2_1"><span class="lab"><font color="Green" title="E36">A87</font></span></a>, <a class="txt" href="#E29:23_2_1"><span class="lab"><font color="Green" title="E37">A88</font></span></a>, <a class="txt" href="#E17:23_2_1"><span class="lab"><font color="Green" title="E29">A80</font></span></a>, <a class="txt" href="#E16:23_2_1"><span class="lab"><font color="Green" title="E28">A79</font></span></a>, <a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E39:23_2_1"/>
<font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c8">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:23_2"/>
<font color="Maroon" title="c1">X</font> is  <a href="conaffm.html#V1" title="CONAFFM:attr.1">satisfying_DES</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conaffm.html#D1" title="CONAFFM:def.1">CONAFFM:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:23"/>
( <font color="Maroon" title="c1">X</font> is  <a href="conaffm.html#V1" title="CONAFFM:attr.1">satisfying_DES</a>  iff  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
