<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">OAS</font> being  <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>  st <font color="Olive" title="b1">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> holds <br/> <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Olive" title="b1">OAS</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</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="c2">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> implies  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">AS</font> =  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font>;<br/>
<a NAME="E2:30"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font></span>)</span><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a9</font>, <font color="Olive" title="b8">b9</font>, <font color="Olive" title="b9">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b3">C</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b8">b9</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b9">c9</font> holds <br/><font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b9</font>,<font color="Olive" title="b9">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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 <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font></span>)</span>  st <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c4">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c5">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c6">C</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b6">c9</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">a9</font>, <font color="Maroon" title="c11">b9</font>, <font color="Maroon" title="c12">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c10">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c11">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c12">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c4">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c5">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c6">C</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c12">c9</font> implies <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">and </span><br/><a NAME="E3:30_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c10">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<font color="Maroon" title="c11">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">and </span><br/><a NAME="E8:30_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c12">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">and </span><br/><a NAME="E9:30_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E10:30_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c5">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E11:30_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c6">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E12:30_1"/><span class="lab"><font color="Green" title="E38">A13</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="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E13:30_1"/><span class="lab"><font color="Green" title="E39">A14</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="c6">C</font>
 <span class="kw">and </span><br/><a NAME="E14:30_1"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c11">b9</font>
 <span class="kw">and </span><br/><a NAME="E15:30_1"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c12">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c14">b1</font> = <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c15">c1</font> = <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c16">a19</font> = <font color="Maroon" title="c10">a9</font>, <font color="Maroon" title="c17">b19</font> = <font color="Maroon" title="c11">b9</font>, <font color="Maroon" title="c18">c19</font> = <font color="Maroon" title="c12">c9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="papdesaf.html#T2" target="_self" title="PAPDESAF:th.2">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">u</font> = <font color="Maroon" title="c13">a1</font>, <font color="Maroon" title="c20">v</font> = <font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c21">w</font> = <font color="Maroon" title="c15">c1</font>, <font color="Maroon" title="c22">u9</font> = <font color="Maroon" title="c16">a19</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="papdesaf.html#T4" target="_self" title="PAPDESAF:th.4">Th4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E42">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font></span><br/><a NAME="E2:30_1_1"/><span class="lab"><font color="Green" title="E44">A19</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c14">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c14">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="papdesaf.html#T3" target="_self" title="PAPDESAF:th.3">Th3</a></span>;<br/>
<a NAME="E3:30_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E9:30_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E43">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="E4:30_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E12:30_1"><span class="lab"><font color="Green" title="E38">A13</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="E3:30_1_1"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c15">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c15">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a9</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="papdesaf.html#T3" target="_self" title="PAPDESAF:th.3">Th3</a></span>;<br/>
<a NAME="E3:30_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E9:30_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E43">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="E4:30_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E13:30_1"><span class="lab"><font color="Green" title="E39">A14</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="E4:30_1_1"/>
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E8:30_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E5:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">c1</font>,<font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></span>;<br/><a NAME="E6:30_1_1"/>
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E7:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c17">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">v99</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>;<br/><span class="kw">set </span><font color="Maroon" title="c24">w99</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">b199</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c26">c199</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="papdesaf.html#T4" target="_self" title="PAPDESAF:th.4">Th4</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">w</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c22">u9</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c22">u9</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c20">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></span>
<br/>.= 
<font color="Maroon" title="c21">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c20">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></span>
;<br/><a NAME="E10:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c21">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T24" title="ANALOAF:th.24">ANALOAF:24</a></span>;<br/><a NAME="E11:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c21">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E12:30_1_1"/>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u9</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c20">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T25" title="ANALOAF:th.25">ANALOAF:25</a></span>;<br/><a NAME="E13:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u9</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E14:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c25">b199</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></span>;<br/><a NAME="E15:30_1_1"/>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c21">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c22">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T25" title="ANALOAF:th.25">ANALOAF:25</a></span>;<br/><a NAME="E16:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c21">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c22">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E17:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c15">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c26">c199</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></span>;<br/><a NAME="E18:30_1_1"/>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u9</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c21">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T25" title="ANALOAF:th.25">ANALOAF:25</a></span>;<br/><a NAME="E19:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u9</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c21">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E20:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A26</font></span>: 
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a19</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">c1</font>,<font color="Maroon" title="c26">c199</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></span>;<br/><a NAME="E21:30_1_1"/>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c22">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T25" title="ANALOAF:th.25">ANALOAF:25</a></span>;<br/><a NAME="E22:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c22">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E23:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A27</font></span>: 
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c14">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c25">b199</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></span>;<br/><a NAME="E24:30_1_1"/>
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c15">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></span>;<br/><a NAME="E25:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A28</font></span>: 
<font color="Maroon" title="c26">c199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="#E5:30_1_1"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E20:30_1_1"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="pasch.html#T12" title="PASCH:th.12">PASCH:12</a></span>;<br/><a NAME="E26:30_1_1"/>
<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c14">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c16">a19</font>,<font color="Maroon" title="c17">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></span>;<br/><a NAME="E27:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">b199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="txt" href="#E23:30_1_1"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="ref" href="pasch.html#T12" title="PASCH:th.12">PASCH:12</a></span>;<br/><a NAME="E28:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c17">b19</font>,<font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E25:30_1_1"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="txt" href="#E11:30_1_1"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E29:30_1_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_1_2"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font></span><br/><a NAME="E2:30_1_2"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_1_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E43">A29</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="E2:30_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A31</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:30_1_2_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:30_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E8:30_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E11:30_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E2:30_1_2_1"><span class="lab"><font color="Green" title="E45">A31</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="E5:30_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E13:30_1"><span class="lab"><font color="Green" title="E39">A14</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="E3:30_1_2"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_1_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E43">A29</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="E2:30_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A32</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:30_1_2_2"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b9</font>
 ; <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="E4:30_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:30_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E2:30_1_2_2"><span class="lab"><font color="Green" title="E45">A32</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="E5:30_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E12:30_1"><span class="lab"><font color="Green" title="E38">A13</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">hence </span><a NAME="E4:30_1_2"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E20:30_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b9</font>,<font color="Maroon" title="c12">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:30"/>
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
