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<div class="add">

<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="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a>  iff <font color="Maroon" title="c1">MS</font> is  <a href="conmetr.html#V5" title="CONMETR:attr.5">satisfying_TDES</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="E20">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c1">MS</font> is  <a href="conmetr.html#V5" title="CONMETR:attr.5">satisfying_TDES</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="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">K</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="Maroon" title="c3">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; not <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b6">b9</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> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">K</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="c4">o</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c9">b9</font>, <font color="Maroon" title="c10">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="c3">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b9</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">K</font> implies <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c3">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:21_1_1"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_1"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E4:21_1_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E5:21_1_1"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E6:21_1_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E7:21_1_1"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E8:21_1_1"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>
 <span class="kw">and </span><br/><a NAME="E9:21_1_1"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b9</font>
 <span class="kw">and </span><br/><a NAME="E10:21_1_1"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 <span class="kw">and </span><br/><a NAME="E11:21_1_1"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E12:21_1_1"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">K</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="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">o1</font> = <font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c13">b1</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c14">c1</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c15">a19</font> = <font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c16">b19</font> = <font color="Maroon" title="c9">b9</font>, <font color="Maroon" title="c17">c19</font> = <font color="Maroon" title="c10">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/><a NAME="E14:21_1_1"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
( <font color="Maroon" title="c11">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b1</font> &amp; <font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c14">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">a19</font>,<font color="Maroon" title="c17">c19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E11:21_1_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E15:21_1_1"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">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="E2:21_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="c4">o</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:21_1_1"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c18">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>;<br/>
<span class="kw">assume </span><a NAME="E1:21_1_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E3:21_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E4:21_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D5" title="AFF_1:def.5">AFF_1:def 5</a></span>;<br/>
<a NAME="E5:21_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E17:21_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/><a NAME="E18:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E19:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c13">b1</font>,<font color="Maroon" title="c12">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c14">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>;<br/><a NAME="E20:21_1_1"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c15">a19</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c13">b1</font>,<font color="Maroon" title="c16">b19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E9:21_1_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E21:21_1_1"/>
<font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c13">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">a19</font>,<font color="Maroon" title="c16">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E22:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<font color="Maroon" title="c13">b1</font>,<font color="Maroon" title="c12">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">b19</font>,<font color="Maroon" title="c15">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></span>;<br/><a NAME="E23:21_1_1"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E24:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A23</font></span>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c14">c1</font>,<font color="Maroon" title="c17">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/><div><span class="lab"><font color="Green" title="E42">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_1_3"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">o</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="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font></span><br/><div><span class="lab"><font color="Green" title="E44">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_1_3_1"/><span class="lab"><font color="Green" title="E45">A27</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">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="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font></span><br/><a NAME="E2:21_1_1_3_1"/><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="c12">a1</font>,<font color="Maroon" title="c15">a19</font>,<font color="Maroon" title="c14">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_3_1_1"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c15">a19</font>,<font color="Maroon" title="c14">c1</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="E2:21_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A29</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c7">c</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="E3:21_1_1_3_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c4">o</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_1_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E1:21_1_1_3_1"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="txt" href="#E2:21_1_1_3_1_1"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:21_1_1_3_1"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c15">a19</font>,<font color="Maroon" title="c13">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_3_1_2"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c15">a19</font>,<font color="Maroon" title="c13">b1</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="E2:21_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A31</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c6">b</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="E3:21_1_1_3_1_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c4">o</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c5">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_1_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_1_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E1:21_1_1_3_1"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="txt" href="#E2:21_1_1_3_1_2"><span class="lab"><font color="Green" title="E48">A31</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:21_1_1_3_1"/>
( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_1_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E5:21_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<font color="Maroon" title="c11">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E1:21_1_1_3"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></span>;<br/><a NAME="E6:21_1_1_3_1"/>
( <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c8">a9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E16:21_1_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E7:21_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E1:21_1_1_3"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></span>;<br/><a NAME="E8:21_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">b1</font>,<font color="Maroon" title="c14">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">b19</font>,<font color="Maroon" title="c17">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E20:21_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E24:21_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="txt" href="#E22:21_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E19:21_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="txt" href="#E14:21_1_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E1:21_1_1_3"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="txt" href="#E5:21_1_1_3_1"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="txt" href="#E2:21_1_1_3_1"><span class="lab"><font color="Green" title="E46">A28</font></span></a>, <a class="txt" href="#E3:21_1_1_3_1"><span class="lab"><font color="Green" title="E47">A30</font></span></a>, <a class="ref" href="conmetr.html#D5" title="CONMETR:def.5">CONMETR:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:21_1_1_3_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_1_1_3_2"/><span class="lab"><font color="Green" title="E45">A33</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/><span class="kw">assume </span><a NAME="E2:21_1_1_3_2"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font></span><br/><a NAME="E3:21_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></span>;<br/><a NAME="E4:21_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E11:21_1_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E23:21_1_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E1:21_1_1_3_2"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</a></span>;<br/><a NAME="E5:21_1_1_3_2"/><span class="lab"><font color="Green" title="E48">A36</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/><a NAME="E6:21_1_1_3_2"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_3_2"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></span>;<br/><a NAME="E7:21_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E9:21_1_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E16:21_1_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E5:21_1_1_3_2"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_1_1_3_2"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1_3_2"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:21_1_1_3"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_3"><span class="lab"><font color="Green" title="E44">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_1_4"/>
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</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="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font></span><br/><a NAME="E2:21_1_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_1_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E11:21_1_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E23:21_1_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E16:21_1_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_1_1_4"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</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="E27:21_1_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21_1_1"><span class="lab"><font color="Green" title="E42">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:21_1"/><span class="kw">then </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#V7" title="AFF_2:attr.7">Moufangian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#D7" title="AFF_2:def.7">AFF_2:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_1"/>
<font color="Maroon" title="c1">MS</font> is  <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclmetr.html#D5" target="_self" title="EUCLMETR:def.5">Def5</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="21_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_2"/>
<font color="Maroon" title="c1">MS</font> is  <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</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="conmetr.html#V5" title="CONMETR:attr.5">satisfying_TDES</a> </span><br/><a NAME="E2:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A37</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#V7" title="AFF_2:attr.7">Moufangian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclmetr.html#D5" target="_self" title="EUCLMETR:def.5">Def5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="c6">b</font>,<font color="Maroon" title="c7">b1</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="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">c</font> &amp; <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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_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="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E2:21_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="E3:21_2_1"/><span class="lab"><font color="Green" title="E22">A38</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="E4:21_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="E5:21_2_1"/><span class="lab"><font color="Green" title="E23">A39</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="E6:21_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="E7:21_2_1"/><span class="lab"><font color="Green" title="E24">A40</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="E8:21_2_1"/><span class="lab"><font color="Green" title="E25">A41</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="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E9:21_2_1"/><span class="lab"><font color="Green" title="E26">A42</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="E10:21_2_1"/><span class="lab"><font color="Green" title="E27">A43</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="E11:21_2_1"/><span class="lab"><font color="Green" title="E28">A44</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="E12:21_2_1"/><span class="lab"><font color="Green" title="E29">A45</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>
 <span class="kw">and </span><br/><a NAME="E13:21_2_1"/><span class="lab"><font color="Green" title="E30">A46</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="c3">o</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E14:21_2_1"/><span class="lab"><font color="Green" title="E31">A47</font></span>: 
<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>
 ; <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="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></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">a19</font> = <font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c13">b9</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c14">b19</font> = <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c15">c9</font> = <font color="Maroon" title="c8">c</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/><span class="kw">set </span><font color="Maroon" title="c17">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>;<br/><a NAME="E16:21_2_1"/><span class="lab"><font color="Green" title="E32">A48</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2_1"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E17:21_2_1"/>
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c13">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_2_1"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E18:21_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c13">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2_1"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="ref" href="aff_1.html#D4" title="AFF_1:def.4">AFF_1:def 4</a></span>;<br/><a NAME="E19:21_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A49</font></span>: 
<font color="Maroon" title="c13">b9</font>,<font color="Maroon" title="c11">a9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/><a NAME="E20:21_2_1"/>
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c13">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c14">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_2_1"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E21:21_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A50</font></span>: 
<font color="Maroon" title="c13">b9</font>,<font color="Maroon" title="c11">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">b19</font>,<font color="Maroon" title="c12">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E22:21_2_1"/><span class="lab"><font color="Green" title="E35">A51</font></span>: 
<font color="Maroon" title="c15">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E23:21_2_1"/><span class="lab"><font color="Green" title="E36">A52</font></span>: 
(  <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="c12">a19</font> &amp; <font color="Maroon" title="c13">b9</font>,<font color="Maroon" title="c15">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">b19</font>,<font color="Maroon" title="c16">c19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_2_1"><span class="lab"><font color="Green" title="E26">A42</font></span></a>, <a class="txt" href="#E14:21_2_1"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E24:21_2_1"/><span class="lab"><font color="Green" title="E37">A53</font></span>: 
<font color="Maroon" title="c13">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2_1"><span class="lab"><font color="Green" title="E24">A40</font></span></a>, <a class="ref" href="euclmetr.html#T1" target="_self" title="EUCLMETR:th.1">Th1</a></span>;<br/><a NAME="E25:21_2_1"/><span class="lab"><font color="Green" title="E38">A54</font></span>: 
<font color="Maroon" title="c10">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E26:21_2_1"/><span class="lab"><font color="Green" title="E39">A55</font></span>: 
 <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">b9</font>,<font color="Maroon" title="c14">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_2_1"><span class="lab"><font color="Green" title="E27">A43</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E27:21_2_1"/><span class="lab"><font color="Green" title="E40">A56</font></span>: 
not <font color="Maroon" title="c13">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_2_1_1"/><span class="lab"><font color="Green" title="E41">A57</font></span>: 
<font color="Maroon" title="c13">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_2_1"><span class="lab"><font color="Green" title="E22">A38</font></span></a>, <a class="txt" href="#E16:21_2_1"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E25:21_2_1"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E26:21_2_1"><span class="lab"><font color="Green" title="E39">A55</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E3:21_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="c13">b9</font>,<font color="Maroon" title="c14">b19</font>,<font color="Maroon" title="c15">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_2_1"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E22:21_2_1"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E1:21_2_1_1"><span class="lab"><font color="Green" title="E41">A57</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="E4:21_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2_1"><span class="lab"><font color="Green" title="E25">A41</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="E28:21_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="c15">c9</font>,<font color="Maroon" title="c16">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_2_1"><span class="lab"><font color="Green" title="E28">A44</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/><a NAME="E29:21_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">c19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2_1"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="txt" href="#E16:21_2_1"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E25:21_2_1"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E22:21_2_1"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E30:21_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c15">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c16">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="txt" href="#E5:21_2_1"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="txt" href="#E16:21_2_1"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E25:21_2_1"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E22:21_2_1"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E26:21_2_1"><span class="lab"><font color="Green" title="E39">A55</font></span></a>, <a class="txt" href="#E27:21_2_1"><span class="lab"><font color="Green" title="E40">A56</font></span></a>, <a class="txt" href="#E19:21_2_1"><span class="lab"><font color="Green" title="E33">A49</font></span></a>, <a class="txt" href="#E21:21_2_1"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="txt" href="#E23:21_2_1"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E24:21_2_1"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="aff_2.html#D7" title="AFF_2:def.7">AFF_2:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E31:21_2_1"/>
<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>
 <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="E4:21_2"/>
<font color="Maroon" title="c1">MS</font> is  <a href="conmetr.html#V5" title="CONMETR:attr.5">satisfying_TDES</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr.html#D5" title="CONMETR:def.5">CONMETR:def 5</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="E3:21"/>
( <font color="Maroon" title="c1">MS</font> is  <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a>  iff <font color="Maroon" title="c1">MS</font> is  <a href="conmetr.html#V5" title="CONMETR:attr.5">satisfying_TDES</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E20">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/>


</div>
