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<span class="kw">let </span><font color="Maroon" title="c1">MS</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">MS</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">MS</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">AS</font> =  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font>;<br/>
<div><span class="lab"><font color="Green" title="E19">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c1">MS</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</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="19_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">C</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">MS</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</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">MS</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="c3">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="c4">P</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</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="c3">A</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c4">P</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c5">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</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">a9</font>,<font color="Olive" title="b6">b9</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">a9</font>,<font color="Olive" title="b7">c9</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">b9</font>,<font color="Olive" title="b7">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">a9</font>, <font color="Maroon" title="c11">b9</font>, <font color="Maroon" title="c12">c9</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">MS</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c10">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c11">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c12">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c5">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c12">c9</font> implies <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_1_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E3:19_1_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">and </span><br/><a NAME="E4:19_1_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">and </span><br/><a NAME="E5:19_1_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">and </span><br/><a NAME="E6:19_1_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">and </span><br/><a NAME="E7:19_1_1"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E8:19_1_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c10">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E9:19_1_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E10:19_1_1"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c11">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E11:19_1_1"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">and </span><br/><a NAME="E12:19_1_1"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c12">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">and </span><br/><a NAME="E13:19_1_1"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<font color="Maroon" title="c3">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="E14:19_1_1"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
<font color="Maroon" title="c4">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E15:19_1_1"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
<font color="Maroon" title="c5">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E16:19_1_1"/><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E17:19_1_1"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">and </span><br/><a NAME="E18:19_1_1"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</font>
 <span class="kw">and </span><br/><a NAME="E19:19_1_1"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c12">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c13">o1</font> = <font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c16">c1</font> = <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c17">a19</font> = <font color="Maroon" title="c10">a9</font>, <font color="Maroon" title="c18">b19</font> = <font color="Maroon" title="c11">b9</font>, <font color="Maroon" title="c19">c19</font> = <font color="Maroon" title="c12">c9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:19_1_1_1"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a9</font>
 <span class="kw">and </span><br/><a NAME="E3:19_1_1_1"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font></span><br/><a NAME="E4:19_1_1_1"/><span class="lab"><font color="Green" title="E42">A24</font></span>: 
( <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a19</font>,<font color="Maroon" title="c18">b19</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c16">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a19</font>,<font color="Maroon" title="c19">c19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:19_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E19:19_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E5:19_1_1_1"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E5:19_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E8:19_1_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E9:19_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E14:19_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E16:19_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E18:19_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="aff_4.html#T9" title="AFF_4:th.9">AFF_4:9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_1_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c7">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:19_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:19_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E10:19_1_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E14:19_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E5:19_1_1_1"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E14:19_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E16:19_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:19_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A26</font></span>: 
not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c18">b19</font>,<font color="Maroon" title="c14">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E8:19_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E8:19_1_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E9:19_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A27</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c17">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_1_1_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:19_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E8:19_1_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E11:19_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E15:19_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E17:19_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:19_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A28</font></span>: 
not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c17">a19</font>,<font color="Maroon" title="c16">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E12:19_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E9:19_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E10:19_1_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E14:19_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E13:19_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A29</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c18">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E14:19_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E11:19_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E12:19_1_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E15:19_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E15:19_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A30</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c19">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E16:19_1_1_1"/>
( <font color="Maroon" title="c13">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">b19</font> &amp; <font color="Maroon" title="c13">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">c19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E5:19_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E8:19_1_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E9:19_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E11:19_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E14:19_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E15:19_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E16:19_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E17:19_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="txt" href="#E18:19_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E19:19_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E2:19_1_1_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="aff_4.html#T8" title="AFF_4:th.8">AFF_4:8</a></span>;<br/><a NAME="E17:19_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b19</font>,<font color="Maroon" title="c19">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E5:19_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E2:19_1_1_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="txt" href="#E9:19_1_1_1"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="txt" href="#E13:19_1_1_1"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="txt" href="#E15:19_1_1_1"><span class="lab"><font color="Green" title="E48">A30</font></span></a>, <a class="txt" href="#E4:19_1_1_1"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="txt" href="#E7:19_1_1_1"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="txt" href="#E11:19_1_1_1"><span class="lab"><font color="Green" title="E46">A28</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="E18:19_1_1_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E21:19_1_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E5:19_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E9:19_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E10:19_1_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E11:19_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E12:19_1_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E14:19_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E15:19_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E16:19_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E17:19_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="txt" href="#E18:19_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E19:19_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="ref" href="afproj.html#T50" title="AFPROJ:th.50">AFPROJ:50</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:19_1"/>
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_2"/><span class="lab"><font color="Green" title="E20">A31</font></span>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">MS</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="19_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a1</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c3">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="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font> &amp; not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c4">a</font> &amp; not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c8">c</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">a1</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b1</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c9">c1</font> implies <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c9">c1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_2_1"/><span class="lab"><font color="Green" title="E21">A32</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E2:19_2_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a1</font>
 <span class="kw">and </span><br/><a NAME="E3:19_2_1"/><span class="lab"><font color="Green" title="E22">A33</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E4:19_2_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b1</font>
 <span class="kw">and </span><br/><a NAME="E5:19_2_1"/><span class="lab"><font color="Green" title="E23">A34</font></span>: 
<font color="Maroon" title="c3">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="E6:19_2_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 <span class="kw">and </span><br/><a NAME="E7:19_2_1"/><span class="lab"><font color="Green" title="E24">A35</font></span>: 
not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E8:19_2_1"/><span class="lab"><font color="Green" title="E25">A36</font></span>: 
not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E9:19_2_1"/><span class="lab"><font color="Green" title="E26">A37</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">a1</font>
 <span class="kw">and </span><br/><a NAME="E10:19_2_1"/><span class="lab"><font color="Green" title="E27">A38</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b1</font>
 <span class="kw">and </span><br/><a NAME="E11:19_2_1"/><span class="lab"><font color="Green" title="E28">A39</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">c1</font>
 <span class="kw">and </span><br/><a NAME="E12:19_2_1"/><span class="lab"><font color="Green" title="E29">A40</font></span>: 
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c9">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c9">c1</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">o9</font> = <font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c11">a9</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c12">b9</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c13">c9</font> = <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c14">a19</font> = <font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c15">b19</font> = <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c16">c19</font> = <font color="Maroon" title="c9">c1</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">MS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></span>;<br/><a NAME="E14:19_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c14">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:19_2_1"><span class="lab"><font color="Green" title="E26">A37</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">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">MS</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E16:19_2_1"/><span class="lab"><font color="Green" title="E30">A41</font></span>: 
<font color="Maroon" title="c17">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="E17:19_2_1"/><span class="lab"><font color="Green" title="E31">A42</font></span>: 
<font color="Maroon" title="c10">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A</font>
 <span class="kw">and </span><br/><a NAME="E18:19_2_1"/><span class="lab"><font color="Green" title="E32">A43</font></span>: 
<font color="Maroon" title="c11">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A</font>
 <span class="kw">and </span><br/><a NAME="E19:19_2_1"/><span class="lab"><font color="Green" title="E33">A44</font></span>: 
<font color="Maroon" title="c14">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E20:19_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c13">c9</font>,<font color="Maroon" title="c16">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:19_2_1"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">C</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">MS</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E22:19_2_1"/><span class="lab"><font color="Green" title="E34">A45</font></span>: 
( <font color="Maroon" title="c18">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c10">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">C</font> )
 <span class="kw">and </span><br/><a NAME="E23:19_2_1"/><span class="lab"><font color="Green" title="E35">A46</font></span>: 
<font color="Maroon" title="c13">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">C</font>
 <span class="kw">and </span><br/><a NAME="E24:19_2_1"/><span class="lab"><font color="Green" title="E36">A47</font></span>: 
<font color="Maroon" title="c16">c19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E25:19_2_1"/><span class="lab"><font color="Green" title="E37">A48</font></span>: 
<font color="Maroon" title="c17">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_2_1_1"/>
<font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_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="c11">a9</font>,<font color="Maroon" title="c14">a19</font>,<font color="Maroon" title="c13">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:19_2_1"><span class="lab"><font color="Green" title="E30">A41</font></span></a>, <a class="txt" href="#E18:19_2_1"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="txt" href="#E19:19_2_1"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="txt" href="#E23:19_2_1"><span class="lab"><font color="Green" title="E35">A46</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:19_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:19_2_1"><span class="lab"><font color="Green" title="E25">A36</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E26:19_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c15">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19_2_1"><span class="lab"><font color="Green" title="E27">A38</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">P</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">MS</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E28:19_2_1"/><span class="lab"><font color="Green" title="E38">A49</font></span>: 
<font color="Maroon" title="c19">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E29:19_2_1"/><span class="lab"><font color="Green" title="E39">A50</font></span>: 
<font color="Maroon" title="c10">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">P</font>
 <span class="kw">and </span><br/><a NAME="E30:19_2_1"/><span class="lab"><font color="Green" title="E40">A51</font></span>: 
( <font color="Maroon" title="c12">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">P</font> &amp; <font color="Maroon" title="c15">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E31:19_2_1"/><span class="lab"><font color="Green" title="E41">A52</font></span>: 
<font color="Maroon" title="c17">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_2_1_2"/>
<font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_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="c12">b9</font>,<font color="Maroon" title="c15">b19</font>,<font color="Maroon" title="c11">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:19_2_1"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="txt" href="#E28:19_2_1"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="txt" href="#E30:19_2_1"><span class="lab"><font color="Green" title="E40">A51</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:19_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_2_1"><span class="lab"><font color="Green" title="E24">A35</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E32:19_2_1"/>
( <font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a19</font>,<font color="Maroon" title="c15">b19</font> &amp; <font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c13">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a19</font>,<font color="Maroon" title="c16">c19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:19_2_1"><span class="lab"><font color="Green" title="E29">A40</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E33:19_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c13">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">b19</font>,<font color="Maroon" title="c16">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E20">A31</font></span></a>, <a class="txt" href="#E1:19_2_1"><span class="lab"><font color="Green" title="E21">A32</font></span></a>, <a class="txt" href="#E3:19_2_1"><span class="lab"><font color="Green" title="E22">A33</font></span></a>, <a class="txt" href="#E5:19_2_1"><span class="lab"><font color="Green" title="E23">A34</font></span></a>, <a class="txt" href="#E16:19_2_1"><span class="lab"><font color="Green" title="E30">A41</font></span></a>, <a class="txt" href="#E17:19_2_1"><span class="lab"><font color="Green" title="E31">A42</font></span></a>, <a class="txt" href="#E18:19_2_1"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="txt" href="#E19:19_2_1"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="txt" href="#E28:19_2_1"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="txt" href="#E29:19_2_1"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="txt" href="#E30:19_2_1"><span class="lab"><font color="Green" title="E40">A51</font></span></a>, <a class="txt" href="#E22:19_2_1"><span class="lab"><font color="Green" title="E34">A45</font></span></a>, <a class="txt" href="#E23:19_2_1"><span class="lab"><font color="Green" title="E35">A46</font></span></a>, <a class="txt" href="#E24:19_2_1"><span class="lab"><font color="Green" title="E36">A47</font></span></a>, <a class="txt" href="#E31:19_2_1"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="txt" href="#E25:19_2_1"><span class="lab"><font color="Green" title="E37">A48</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="E34:19_2_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c9">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:19_2"/>
<font color="Maroon" title="c1">MS</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:19"/>
( <font color="Maroon" title="c1">MS</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">MS</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="#E1:19"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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