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<b>let </b><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a> implies <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V8" title="AFF_2:attr.8">satisfying_TDES_1</a> )</span><br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V8" title="AFF_2:attr.8">satisfying_TDES_1</a></span><br/>

<b>thus </b><a NAME="E2:17"/>
<font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V8" title="AFF_2:attr.8">satisfying_TDES_1</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D8" target="_self" title="AFF_2:def.8">AFF_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <font color="Olive" title="b6">b'</font>, <font color="Olive" title="b7">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">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="c2">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="c2">K</font> &amp; <font color="Olive" title="b7">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="c2">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">a'</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">a'</font>,<font color="Olive" title="b6">b'</font> &amp; <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">b'</font>,<font color="Olive" title="b7">c'</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">a'</font>,<font color="Olive" title="b7">c'</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="c2">K</font> holds <br/> <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">b'</font></span><br/><b>let </b><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">b'</font>, <font color="Maroon" title="c9">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c9">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</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">c</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c9">c'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c9">c'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> implies  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b'</font> )</span><br/>















<b>assume </b><b>that </b><br/><a NAME="E1:17_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 <b>and </b><br/><a NAME="E2:17_1"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c9">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> )
 <b>and </b><br/><a NAME="E3:17_1"/><i><font color="Green" title="E18">A4</font></i>: 
not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E4:17_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c9">c'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c9">c'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b'</font></span><br/>

<b>assume </b><a NAME="E5:17_1"/><i><font color="Green" title="E20">A6</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:17_1"/><i><font color="Green" title="E21">A7</font></i>: 
not <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></i>;<br/>
<a NAME="E7:17_1"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></i>;<br/>
<a NAME="E8:17_1"/><i><font color="Green" title="E23">A9</font></i>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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="c4">a</font> )
 
<b>by </b><i><a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E5:17_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>;<br/>
<b>set </b><font color="Maroon" title="c11">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>;<br/>
<a NAME="E9:17_1"/><i><font color="Green" title="E24">A10</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">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="c3">o</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c3">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="c3">o</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="txt" href="#E8:17_1"><i><font color="Green" title="E23">A9</font></i></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#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E10:17_1"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E12:17_1"/><i><font color="Green" title="E26">A12</font></i>: 
( <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">P</font> )
 <b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E13:17_1"/><i><font color="Green" title="E27">A13</font></i>: 
( <font color="Maroon" title="c12">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c12">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E14:17_1"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:17_1_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E6:17_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="#E9:17_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c13">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E16:17_1"/><i><font color="Green" title="E28">A14</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font> )
 <b>by </b><i><a class="txt" href="#E9:17_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E13:17_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<a NAME="E17:17_1"/><i><font color="Green" title="E29">A15</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="#E9:17_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E16:17_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E18:17_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c13">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:17_1_2"/>
<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c13">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="#E16:17_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<b>hence </b><a NAME="E2:17_1_2"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E19:17_1"/><b>then </b>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c9">c'</font>
 
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E17:17_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="aff_2.html#D7" target="_self" title="AFF_2:def.7">Def7</a></i>;<br/>
<a NAME="E20:17_1"/><b>then </b>
<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="c13">x</font>,<font color="Maroon" title="c9">c'</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E7:17_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E21:17_1"/><b>then </b>
<font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c8">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E22:17_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E23:17_1"/><b>then </b><i><font color="Green" title="E30">A16</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c9">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E24:17_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c12">P</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E25:17_1"/><b>then </b>
<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c12">P</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<a NAME="E26:17_1"/><b>then </b><i><font color="Green" title="E31">A17</font></i>: 
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font>
 
<b>by </b><i><a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="#E13:17_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<a NAME="E27:17_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c8">b'</font> )
 
<b>by </b><i><a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="#E13:17_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="#E16:17_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E28:17_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c7">a'</font>
 
<b>by </b><i><a class="txt" href="#E5:17_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="#E17:17_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="#E23:17_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E29:17_1"/><b>then </b>
<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="c8">b'</font>,<font color="Maroon" title="c7">a'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E30:17_1"/><b>then </b><i><font color="Green" title="E32">A18</font></i>: 
<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="c7">a'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E31:17_1"/><i><font color="Green" title="E33">A19</font></i>: 
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_3"/><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:17_1_3"/><b>then </b>
<font color="Maroon" title="c12">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="#E26:17_1"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E3:17_1_3"/><b>then </b><i><font color="Green" title="E35">A21</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font>
 
<b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E9:17_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E10:17_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E12:17_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E4:17_1_3"/>
<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="c6">c</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E5:17_1_3"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font>
 
<b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E1:17_1_3"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="#E3:17_1_3"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:17_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E32:17_1"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_4"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:17_1_4"/><b>then </b><i><font color="Green" title="E36">A24</font></i>: 
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c'</font> )
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_4_1"/>
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_1_4_1"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font>
 <b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E9:17_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E10:17_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E1:17_1_4"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E3:17_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:17_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:17_1_4"/><b>then </b>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="#E2:17_1_4"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E5:17_1_4"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:17_1_4"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E7:17_1_4"/><b>then </b>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E8:17_1_4"/><b>then </b>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E9:17_1_4"/><b>then </b>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<a NAME="E10:17_1_4"/><b>then </b>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></i>;<br/>
<b>hence </b><a NAME="E11:17_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E33:17_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E30:17_1"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="#E31:17_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E34:17_1"/><b>then </b>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E32:17_1"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E35:17_1"/><b>then </b>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<a NAME="E36:17_1"/><b>then </b>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></i>;<br/>
<b>hence </b><a NAME="E37:17_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E6:17_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
