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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#V2" title="AFF_2:attr.2">Pappian</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</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#V13" title="AFF_2:attr.13">satisfying_pap</a> </span><br/>

<span class="kw">thus </span><a NAME="E2:37"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</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="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">M</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#D13" target="_self" title="AFF_2:def.13">AFF_2:def 13</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">N</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="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a9</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b9</font> holds <br/><font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</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">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</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">N</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">a9</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">b9</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b6">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">a9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a9</font>, <font color="Maroon" title="c8">b9</font>, <font color="Maroon" title="c9">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">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> implies <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:37_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:37_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E4:37_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E5:37_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E6:37_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E7:37_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E8:37_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E9:37_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E10:37_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E11:37_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">and </span><br/><a NAME="E12:37_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font></span><br/>

<a NAME="E13:37_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E25">A9</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="c10">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>;<br/>
<a NAME="E14:37_1"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E15:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</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/>
<span class="kw">assume </span><a NAME="E16:37_1"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
not <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</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/>

<div><span class="lab"><font color="Green" title="E34">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_1"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:37_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E28">A12</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="E3:37_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c4">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="E4:37_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</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="E5:37_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E16:37_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E35">A19</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="lab"><font color="Green" title="E35">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_2"/>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</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="E2:37_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E3:37_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:37_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">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="E5:37_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E17:37_1"><span class="lab"><font color="Green" title="E34">A18</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="E6:37_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E25">A9</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="lab"><font color="Green" title="E36">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_3"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="E2:37_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E29">A13</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="E3:37_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c5">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="E4:37_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c9</font> or <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</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="E5:37_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E16:37_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E1:37_1_3"><span class="lab"><font color="Green" title="E37">A22</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="lab"><font color="Green" title="E37">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_4"/>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="E2:37_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E3:37_1_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:37_1_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</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="E5:37_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E19:37_1"><span class="lab"><font color="Green" title="E36">A21</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="E6:37_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</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="E21:37_1"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E22:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">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="E23:37_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">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="#E21:37_1"><span class="lab"><font color="Green" title="E38">A24</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="c12">T</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="E25:37_1"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font>
 <span class="kw">and </span><br/><a NAME="E26:37_1"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">T</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="E27:37_1"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c12">T</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="#E26:37_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<a NAME="E28:37_1"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c9">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="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:37_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E29:37_1"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
( <font color="Maroon" title="c6">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="c6">c</font>,<font color="Maroon" title="c8">b9</font> &amp; <font color="Maroon" title="c8">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="c6">c</font>,<font color="Maroon" title="c8">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:37_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E30:37_1"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1_5"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">T</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="E2:37_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</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="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:37_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:37_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E28:37_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E29:37_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E4:37_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E14:37_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E5:37_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c4">a</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="E6:37_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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="E7:37_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</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="E8:37_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E17:37_1"><span class="lab"><font color="Green" title="E34">A18</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="E9:37_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E32:37_1"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">and </span><br/><a NAME="E33:37_1"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E27:37_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font>;<br/>
<a NAME="E34:37_1"/><span class="lab"><font color="Green" title="E47">A33</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:37_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">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E29:37_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E32:37_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E3:37_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E19:37_1"><span class="lab"><font color="Green" title="E36">A21</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="E4:37_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</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="E35:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A34</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</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="E36:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A35</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E37:37_1"/><span class="lab"><font color="Green" title="E50">A36</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</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="#E34:37_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E38:37_1"/><span class="lab"><font color="Green" title="E51">A37</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:37_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E39:37_1"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1_7"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</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="E2:37_1_7"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:37_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E38:37_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E4:37_1_7"/><span class="kw">then </span>
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font> or <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E15:37_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E29:37_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E32:37_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E33:37_1"><span class="lab"><font color="Green" title="E46">A32</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="E5:37_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E16:37_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E22:37_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E28:37_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E25:37_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E26:37_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">o</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="E41:37_1"/><span class="lab"><font color="Green" title="E52">A38</font></span>: 
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E42:37_1"/><span class="lab"><font color="Green" title="E53">A39</font></span>: 
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">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="E43:37_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E29:37_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E32:37_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E44:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A40</font></span>: 
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c13">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="E45:37_1"/><span class="lab"><font color="Green" title="E55">A41</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1_8"/>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="E2:37_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c7">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:37_1"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:37_1_8"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E4:37_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E18:37_1"><span class="lab"><font color="Green" title="E35">A20</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="E5:37_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</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="lab"><font color="Green" title="E56">A42</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_9"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="E2:37_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E25:37_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E27:37_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E33:37_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E42:37_1"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="txt" href="#E45:37_1"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E3:37_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c3">N</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="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E28:37_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E26:37_1"><span class="lab"><font color="Green" title="E41">A27</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="E4:37_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E22:37_1"><span class="lab"><font color="Green" title="E39">A25</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="E47:37_1"/><span class="lab"><font color="Green" title="E57">A43</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E28:37_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E25:37_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E26:37_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E33:37_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_10"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a9</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="E2:37_1_10"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E38:37_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E41:37_1"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E3:37_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c13">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="E4:37_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c7">a9</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="E5:37_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E13:37_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E6:37_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:37_1"><span class="lab"><font color="Green" title="E57">A43</font></span></a>, <a class="txt" href="#E45:37_1"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E7:37_1_10"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">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="E8:37_1_10"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c4">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="E9:37_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E20:37_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E10:37_1_10"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</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="E49:37_1"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:37_1"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E50:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A45</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E14:37_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E51:37_1"/><span class="lab"><font color="Green" title="E60">A46</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E52:37_1"/>
not <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1_11"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:37_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E51:37_1"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/>
<a NAME="E3:37_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</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="c13">x</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<a NAME="E4:37_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:37_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E27:37_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E33:37_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E38:37_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E45:37_1"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E5:37_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E28:37_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E25:37_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E26:37_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E6:37_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E13:37_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E7:37_1_11"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">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="E8:37_1_11"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c4">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="E9:37_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E20:37_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:37_1_11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">y</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="E54:37_1"/><span class="lab"><font color="Green" title="E61">A47</font></span>: 
<font color="Maroon" title="c16">y</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="c13">x</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E55:37_1"/><span class="lab"><font color="Green" title="E62">A48</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></span>;<br/>
<a NAME="E56:37_1"/><span class="lab"><font color="Green" title="E63">A49</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c4">a</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="E57:37_1"/><span class="lab"><font color="Green" title="E64">A50</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1_12"/>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="E2:37_1_12"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E28:37_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E25:37_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E26:37_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E33:37_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E3:37_1_12"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E18:37_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/>
<a NAME="E4:37_1_12"/><span class="kw">then </span>
<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c4">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/>
<a NAME="E5:37_1_12"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</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="E6:37_1_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</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="lab"><font color="Green" title="E65">A51</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_13"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_13"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</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="E2:37_1_13"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E38:37_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E41:37_1"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E3:37_1_13"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c13">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E4:37_1_13"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">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="E5:37_1_13"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:37_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="txt" href="#E57:37_1"><span class="lab"><font color="Green" title="E64">A50</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E6:37_1_13"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c8">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="E7:37_1_13"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c5">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="E8:37_1_13"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E20:37_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E9:37_1_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</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="lab"><font color="Green" title="E66">A52</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_14"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_14"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</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="E2:37_1_14"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:37_1"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E3:37_1_14"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E42:37_1"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="txt" href="#E58:37_1"><span class="lab"><font color="Green" title="E65">A51</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="E4:37_1_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</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="E60:37_1"/><span class="lab"><font color="Green" title="E67">A53</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E68">A54</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_15"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_15"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="E2:37_1_15"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</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="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E14:37_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E60:37_1"><span class="lab"><font color="Green" title="E67">A53</font></span></a>, <a class="txt" href="#E15:37_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E29:37_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E41:37_1"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/><a NAME="E3:37_1_15"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:37_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E38:37_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E4:37_1_15"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E29:37_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E5:37_1_15"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E19:37_1"><span class="lab"><font color="Green" title="E36">A21</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="E6:37_1_15"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</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="E62:37_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c16">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:37_1"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E63:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c16">y</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="E64:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c8">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="E65:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A55</font></span>: 
<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E51:37_1"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E66:37_1"/>
<font color="Maroon" title="c15">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E42:37_1"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E67:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E38:37_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E41:37_1"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="txt" href="#E42:37_1"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="txt" href="#E50:37_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="txt" href="#E54:37_1"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E65:37_1"><span class="lab"><font color="Green" title="E69">A55</font></span></a>, <a class="txt" href="#E36:37_1"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E58:37_1"><span class="lab"><font color="Green" title="E65">A51</font></span></a>, <a class="txt" href="#E48:37_1"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="txt" href="#E61:37_1"><span class="lab"><font color="Green" title="E68">A54</font></span></a>, <a class="txt" href="#E46:37_1"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E59:37_1"><span class="lab"><font color="Green" title="E66">A52</font></span></a>, <a class="ref" href="aff_2.html#D2" target="_self" title="AFF_2:def.2">Def2</a></span>;<br/>
<a NAME="E68:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:37_1"><span class="lab"><font color="Green" title="E57">A43</font></span></a>, <a class="txt" href="#E45:37_1"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E69:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c16">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c4">a</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="E70:37_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c4">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="E71:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A56</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c9">c9</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="E72:37_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:37_1"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E73:37_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E20:37_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E54:37_1"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E71:37_1"><span class="lab"><font color="Green" title="E70">A56</font></span></a>, <a class="txt" href="#E56:37_1"><span class="lab"><font color="Green" title="E63">A49</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E74:37_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:37_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E17:37_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E37:37_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E35:37_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</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="E75:37_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:37_1"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="txt" href="#E42:37_1"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="txt" href="#E36:37_1"><span class="lab"><font color="Green" title="E49">A35</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>
