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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E16">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</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">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> </span><br/><span class="kw">thus </span><a NAME="E2:35_1"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aff_2.html#D11" target="_self" title="AFF_2:def.11">AFF_2:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a9</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">c9</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">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b8">c9</font> holds <br/><font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b8">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font>, <font color="Olive" title="b6">c9</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">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b6">c9</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font></span><br/><span class="kw">let </span><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 <font color="Maroon" title="c1">AP</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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</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> 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="E1:35_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E3:35_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E6:35_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E7:35_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E9:35_1_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E10:35_1_1"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E11:35_1_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E12:35_1_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E13:35_1_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E14:35_1_1"/><span class="lab"><font color="Green" title="E31">A16</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="E15:35_1_1"/><span class="lab"><font color="Green" title="E32">A17</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>
 ; <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="E16:35_1_1"/><span class="lab"><font color="Green" title="E33">A18</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">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:35_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E12:35_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E17:35_1_1"/><span class="lab"><font color="Green" title="E34">A19</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="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E12:35_1_1"><span class="lab"><font color="Green" title="E29">A14</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">set </span><font color="Maroon" title="c11">K</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="c7">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>;<br/>
<a NAME="E18:35_1_1"/><span class="lab"><font color="Green" title="E35">A20</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="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E3:35_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:35_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E13:35_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E19:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<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="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">assume </span><a NAME="E20:35_1_1"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
not <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>
 ; <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="E21:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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>;<br/>
<a NAME="E22:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c9">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="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#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E23:35_1_1"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></span>;<br/>
<a NAME="E24:35_1_1"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
not  <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="c7">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1_1_1"/><span class="lab"><font color="Green" title="E42">A27</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="c6">b</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="E2:35_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</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/>
<a NAME="E3:35_1_1_1"/><span class="kw">then </span>
<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="c6">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E4:35_1_1_1"/><span class="kw">then </span>
<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="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#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:35_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<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="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E6:35_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</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="#E1:35_1_1_1"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E7:35_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <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#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E8:35_1_1_1"/><span class="kw">then </span>
<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="c5">a</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="E9:35_1_1_1"/><span class="kw">then </span>
<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="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:35_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E10:35_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</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">by</span> <span class="lab"><a class="txt" href="#E23:35_1_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E5:35_1_1_1"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E11:35_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="c8">a9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E12:35_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="c9">b9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</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="E13:35_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E14:35_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</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">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:35_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E17:35_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E5:35_1_1_1"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:35_1_1"/><span class="lab"><font color="Green" title="E42">A29</font></span>: 
<font color="Maroon" title="c7">c</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="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E26:35_1_1"/><span class="lab"><font color="Green" title="E43">A30</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">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="#E21:35_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E28:35_1_1"/><span class="lab"><font color="Green" title="E44">A31</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">and </span><br/><a NAME="E29:35_1_1"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E30:35_1_1"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon" title="c10">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="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E31:35_1_1"/><span class="lab"><font color="Green" title="E47">A34</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="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E8:35_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E13:35_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E32:35_1_1"/><span class="lab"><font color="Green" title="E48">A35</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">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="E2:35_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</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">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:35_1_1_2"/><span class="kw">then </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="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E17:35_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:35_1_1_2"/><span class="kw">then </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="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E31:35_1_1"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:35_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E33:35_1_1"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1_1_3"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">M</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="E2:35_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_1"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:35_1_1_3"/><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="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E25:35_1_1"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E30:35_1_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E4:35_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</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="#E15:35_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:35_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E5:35_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">b9</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="E6:35_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E34:35_1_1"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</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="#E18:35_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E35:35_1_1"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<font color="Maroon" title="c13">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_1"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E37:35_1_1"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E38:35_1_1"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:35_1_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E33:35_1_1"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<a NAME="E39:35_1_1"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</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="#E22:35_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E30:35_1_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E28:35_1_1"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="txt" href="#E29:35_1_1"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>;<br/>
<a NAME="E40:35_1_1"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1_1_4"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">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="E2:35_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:35_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E8:35_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E13:35_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E41:35_1_1"/><span class="lab"><font color="Green" title="E56">A43</font></span>: 
<font color="Maroon" title="c14">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="c14">x</font>,<font color="Maroon" title="c10">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="E42:35_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="c7">c</font>,<font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:35_1_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E19:35_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E25:35_1_1"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E43:35_1_1"/><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="c5">a</font>,<font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E44:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A44</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c14">x</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">by</span> <span class="lab"><a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:35_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E45:35_1_1"/><span class="lab"><font color="Green" title="E58">A45</font></span>: 
<font color="Maroon" title="c10">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="c14">x</font>,<font color="Maroon" title="c10">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="E46:35_1_1"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
not  <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="c14">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_1_1_5"/><span class="lab"><font color="Green" title="E60">A47</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="c14">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1_1_5_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:35_1_1_5_1"/><span class="kw">then </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="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E30:35_1_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E28:35_1_1"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="txt" href="#E29:35_1_1"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E3:35_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</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="#E14:35_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E16:35_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:35_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c8">a9</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#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:35_1_1_5_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35_1_1_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1_1"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:35_1_1_5"/>
 <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="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:35_1_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c14">x</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="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E4:35_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A48</font></span>: 
<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E5:35_1_1_5"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:35_1_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E19:35_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E25:35_1_1"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E24:35_1_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E52">A39</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="E6:35_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:35_1_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E19:35_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E28:35_1_1"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="txt" href="#E35:35_1_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E33:35_1_1"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E4:35_1_1_5"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E1:35_1_1_5"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</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="E47:35_1_1"/><span class="lab"><font color="Green" title="E60">A49</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E48:35_1_1"/><span class="lab"><font color="Green" title="E61">A50</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E62">A51</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_6_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:35_1_1_6_1"/><span class="lab"><font color="Green" title="E63">A52</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><span class="kw">assume </span><a NAME="E2:35_1_1_6_1"/><span class="lab"><font color="Green" title="E64">A53</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">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="E3:35_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:35_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E47:35_1_1"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E30:35_1_1"><span class="lab"><font color="Green" title="E46">A33</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="E4:35_1_1_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E30:35_1_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E2:35_1_1_6_1"><span class="lab"><font color="Green" title="E64">A53</font></span></a>, <a class="txt" href="#E1:35_1_1_6_1"><span class="lab"><font color="Green" title="E63">A52</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:35_1_1_6"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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="E3:35_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</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="#E30:35_1_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E29:35_1_1"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E38:35_1_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E4:35_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A54</font></span>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> or <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="#E5:35_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E10:35_1_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E26:35_1_1"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E22:35_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E28:35_1_1"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E5:35_1_1_6"/><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="c6">b</font>,<font color="Maroon" title="c8">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E1:35_1_1_6"><span class="lab"><font color="Green" title="E62">A51</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E6:35_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E7:35_1_1_6"/><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="c8">a9</font>,<font color="Maroon" title="c6">b</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="E8:35_1_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E9:35_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E9:35_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</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="#E1:35_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E12:35_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E15:35_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E10:35_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</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="E11:35_1_1_6"/><span class="kw">then </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> or <font color="Maroon" title="c8">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:35_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E11:35_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:35_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E13:35_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E4:35_1_1_6"><span class="lab"><font color="Green" title="E63">A54</font></span></a>, <a class="txt" href="#E1:35_1_1_6"><span class="lab"><font color="Green" title="E62">A51</font></span></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E49:35_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">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="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E50:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:35_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E9:35_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E10:35_1_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E12:35_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E14:35_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E39:35_1_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E44:35_1_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E45:35_1_1"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E48:35_1_1"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="txt" href="#E46:35_1_1"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E40:35_1_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="aff_2.html#D12" target="_self" title="AFF_2:def.12">Def12</a></span>;<br/>
<a NAME="E51:35_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E52:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E45:35_1_1"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E53:35_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E11:35_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E20:35_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E34:35_1_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E25:35_1_1"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E37:35_1_1"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E39:35_1_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E41:35_1_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E54:35_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E3:35_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E13:35_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E19:35_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_2"/><span class="lab"><font color="Green" title="E17">A55</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</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">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a> </span><br/><span class="kw">thus </span><a NAME="E2:35_2"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aff_2.html#D12" target="_self" title="AFF_2:def.12">AFF_2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a9</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">c9</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">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b8">c9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">c9</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font>, <font color="Olive" title="b6">c9</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">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b6">c9</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c9</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font></span><br/><span class="kw">let </span><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 <font color="Maroon" title="c1">AP</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="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</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="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> &amp; not  <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="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font> implies <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font> )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_2_1"/><span class="lab"><font color="Green" title="E18">A56</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E2:35_2_1"/><span class="lab"><font color="Green" title="E19">A57</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E3:35_2_1"/><span class="lab"><font color="Green" title="E20">A58</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E4:35_2_1"/><span class="lab"><font color="Green" title="E21">A59</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E5:35_2_1"/><span class="lab"><font color="Green" title="E22">A60</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E6:35_2_1"/><span class="lab"><font color="Green" title="E23">A61</font></span>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <span class="kw">and </span><br/><a NAME="E7:35_2_1"/><span class="lab"><font color="Green" title="E24">A62</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E8:35_2_1"/><span class="lab"><font color="Green" title="E25">A63</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E9:35_2_1"/><span class="lab"><font color="Green" title="E26">A64</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E10:35_2_1"/><span class="lab"><font color="Green" title="E27">A65</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E11:35_2_1"/><span class="lab"><font color="Green" title="E28">A66</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E12:35_2_1"/><span class="lab"><font color="Green" title="E29">A67</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="E13:35_2_1"/><span class="lab"><font color="Green" title="E30">A68</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="E14:35_2_1"/><span class="lab"><font color="Green" title="E31">A69</font></span>: 
<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">and </span><br/><a NAME="E15:35_2_1"/><span class="lab"><font color="Green" title="E32">A70</font></span>: 
not  <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="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E16:35_2_1"/><span class="lab"><font color="Green" title="E33">A71</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font></span><br/>

<span class="kw">assume </span><a NAME="E17:35_2_1"/><span class="lab"><font color="Green" title="E34">A72</font></span>: 
not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">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/>

<span class="kw">consider </span><font color="Maroon" title="c11">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E19:35_2_1"/><span class="lab"><font color="Green" title="E35">A73</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font>
 <span class="kw">and </span><br/><a NAME="E20:35_2_1"/><span class="lab"><font color="Green" title="E36">A74</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_2_1"><span class="lab"><font color="Green" title="E24">A62</font></span></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E21:35_2_1"/><span class="lab"><font color="Green" title="E37">A75</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="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_2_1"><span class="lab"><font color="Green" title="E32">A70</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E22:35_2_1"/><span class="lab"><font color="Green" title="E38">A76</font></span>: 
not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_2_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">K</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="E2:35_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_2_1"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></span>;<br/>
<a NAME="E3:35_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A77</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2_1"><span class="lab"><font color="Green" title="E19">A57</font></span></a>, <a class="txt" href="#E7:35_2_1"><span class="lab"><font color="Green" title="E24">A62</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<a NAME="E4:35_2_1_1"/>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_2_1"><span class="lab"><font color="Green" title="E30">A68</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:35_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2_1"><span class="lab"><font color="Green" title="E19">A57</font></span></a>, <a class="txt" href="#E7:35_2_1"><span class="lab"><font color="Green" title="E24">A62</font></span></a>, <a class="txt" href="#E21:35_2_1"><span class="lab"><font color="Green" title="E37">A75</font></span></a>, <a class="txt" href="#E3:35_2_1_1"><span class="lab"><font color="Green" title="E39">A77</font></span></a>, <a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/>
<a NAME="E6:35_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2_1"><span class="lab"><font color="Green" title="E20">A58</font></span></a>, <a class="txt" href="#E7:35_2_1"><span class="lab"><font color="Green" title="E24">A62</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_2_1"><span class="lab"><font color="Green" title="E23">A61</font></span></a>, <a class="txt" href="#E7:35_2_1"><span class="lab"><font color="Green" title="E24">A62</font></span></a>, <a class="txt" href="#E9:35_2_1"><span class="lab"><font color="Green" title="E26">A64</font></span></a>, <a class="txt" href="#E11:35_2_1"><span class="lab"><font color="Green" title="E28">A66</font></span></a>, <a class="txt" href="#E16:35_2_1"><span class="lab"><font color="Green" title="E33">A71</font></span></a>, <a class="txt" href="#E3:35_2_1_1"><span class="lab"><font color="Green" title="E39">A77</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:35_2_1"/><span class="lab"><font color="Green" title="E39">A78</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_2_1_2"/><span class="lab"><font color="Green" title="E40">A79</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K</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="E2:35_2_1_2"/>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_2_1"><span class="lab"><font color="Green" title="E30">A68</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:35_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E20">A58</font></span></a>, <a class="txt" href="#E19:35_2_1"><span class="lab"><font color="Green" title="E35">A73</font></span></a>, <a class="txt" href="#E20:35_2_1"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="txt" href="#E22:35_2_1"><span class="lab"><font color="Green" title="E38">A76</font></span></a>, <a class="txt" href="#E1:35_2_1_2"><span class="lab"><font color="Green" title="E40">A79</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></span>;<br/>
<a NAME="E4:35_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_2_1"><span class="lab"><font color="Green" title="E31">A69</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:35_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font> or <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="c6">b</font>,<font color="Maroon" title="c7">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_2_1"><span class="lab"><font color="Green" title="E29">A67</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E6:35_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <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="c6">b</font>,<font color="Maroon" title="c7">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2_1"><span class="lab"><font color="Green" title="E20">A58</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E7:35_2_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1"><span class="lab"><font color="Green" title="E18">A56</font></span></a>, <a class="txt" href="#E5:35_2_1"><span class="lab"><font color="Green" title="E22">A60</font></span></a>, <a class="txt" href="#E10:35_2_1"><span class="lab"><font color="Green" title="E27">A65</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:35_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_2_1"><span class="lab"><font color="Green" title="E32">A70</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E40">A80</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_2_1_3"/>
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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="E2:35_2_1_3"/><span class="kw">then </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="c5">a</font>,<font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c8">a9</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="#E12:35_2_1"><span class="lab"><font color="Green" title="E29">A67</font></span></a>, <a class="txt" href="#E13:35_2_1"><span class="lab"><font color="Green" title="E30">A68</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1"><span class="lab"><font color="Green" title="E18">A56</font></span></a>, <a class="txt" href="#E3:35_2_1"><span class="lab"><font color="Green" title="E20">A58</font></span></a>, <a class="txt" href="#E5:35_2_1"><span class="lab"><font color="Green" title="E22">A60</font></span></a>, <a class="txt" href="#E10:35_2_1"><span class="lab"><font color="Green" title="E27">A65</font></span></a>, <a class="txt" href="#E15:35_2_1"><span class="lab"><font color="Green" title="E32">A70</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E25:35_2_1"/><span class="lab"><font color="Green" title="E41">A81</font></span>: 
<font color="Maroon" title="c11">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_2_1"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E27:35_2_1"/><span class="lab"><font color="Green" title="E42">A82</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font>
 <span class="kw">and </span><br/><a NAME="E28:35_2_1"/><span class="lab"><font color="Green" title="E43">A83</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="c7">c</font>,<font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_2_1"><span class="lab"><font color="Green" title="E38">A76</font></span></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></span>;<br/>
<a NAME="E29:35_2_1"/>
<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="c5">a</font>,<font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:35_2_1"><span class="lab"><font color="Green" title="E43">A83</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E30:35_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c12">x</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">by</span> <span class="lab"><a class="txt" href="#E13:35_2_1"><span class="lab"><font color="Green" title="E30">A68</font></span></a>, <a class="txt" href="#E21:35_2_1"><span class="lab"><font color="Green" title="E37">A75</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E31:35_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">x</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="#E1:35_2"><span class="lab"><font color="Green" title="E17">A55</font></span></a>, <a class="txt" href="#E1:35_2_1"><span class="lab"><font color="Green" title="E18">A56</font></span></a>, <a class="txt" href="#E2:35_2_1"><span class="lab"><font color="Green" title="E19">A57</font></span></a>, <a class="txt" href="#E3:35_2_1"><span class="lab"><font color="Green" title="E20">A58</font></span></a>, <a class="txt" href="#E4:35_2_1"><span class="lab"><font color="Green" title="E21">A59</font></span></a>, <a class="txt" href="#E5:35_2_1"><span class="lab"><font color="Green" title="E22">A60</font></span></a>, <a class="txt" href="#E7:35_2_1"><span class="lab"><font color="Green" title="E24">A62</font></span></a>, <a class="txt" href="#E8:35_2_1"><span class="lab"><font color="Green" title="E25">A63</font></span></a>, <a class="txt" href="#E10:35_2_1"><span class="lab"><font color="Green" title="E27">A65</font></span></a>, <a class="txt" href="#E12:35_2_1"><span class="lab"><font color="Green" title="E29">A67</font></span></a>, <a class="txt" href="#E19:35_2_1"><span class="lab"><font color="Green" title="E35">A73</font></span></a>, <a class="txt" href="#E20:35_2_1"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="txt" href="#E25:35_2_1"><span class="lab"><font color="Green" title="E41">A81</font></span></a>, <a class="txt" href="#E27:35_2_1"><span class="lab"><font color="Green" title="E42">A82</font></span></a>, <a class="txt" href="#E23:35_2_1"><span class="lab"><font color="Green" title="E39">A78</font></span></a>, <a class="ref" href="aff_2.html#D11" target="_self" title="AFF_2:def.11">Def11</a></span>;<br/>
<a NAME="E32:35_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c9">b9</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="#E14:35_2_1"><span class="lab"><font color="Green" title="E31">A69</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E33:35_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_2_1"><span class="lab"><font color="Green" title="E40">A80</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E34:35_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A84</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</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="E35:35_2_1"/><span class="lab"><font color="Green" title="E45">A85</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</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/>
<a NAME="E36:35_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</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="#E28:35_2_1"><span class="lab"><font color="Green" title="E43">A83</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E37:35_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_2_1"><span class="lab"><font color="Green" title="E32">A70</font></span></a>, <a class="txt" href="#E27:35_2_1"><span class="lab"><font color="Green" title="E42">A82</font></span></a>, <a class="txt" href="#E34:35_2_1"><span class="lab"><font color="Green" title="E44">A84</font></span></a>, <a class="txt" href="#E35:35_2_1"><span class="lab"><font color="Green" title="E45">A85</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:35_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_2_1"><span class="lab"><font color="Green" title="E23">A61</font></span></a>, <a class="txt" href="#E9:35_2_1"><span class="lab"><font color="Green" title="E26">A64</font></span></a>, <a class="txt" href="#E16:35_2_1"><span class="lab"><font color="Green" title="E33">A71</font></span></a>, <a class="txt" href="#E17:35_2_1"><span class="lab"><font color="Green" title="E34">A72</font></span></a>, <a class="txt" href="#E19:35_2_1"><span class="lab"><font color="Green" title="E35">A73</font></span></a>, <a class="txt" href="#E20:35_2_1"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="txt" href="#E25:35_2_1"><span class="lab"><font color="Green" title="E41">A81</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:35"/>
( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E16">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>
