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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E27">A1</font></i>: 
<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>
 ;<br/>

<b>set </b><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="E2:33"/>
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">a'</font>, <font color="Olive" title="b8">b'</font>, <font color="Olive" title="b9">c'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">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="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">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="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">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="b1">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b3">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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">a'</font>,<font color="Olive" title="b8">b'</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">a'</font>,<font color="Olive" title="b9">c'</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">b'</font>,<font color="Olive" title="b9">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><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>;<br/><b>let </b><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">a'</font>, <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c12">c'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1"/><i><font color="Green" title="E28">A2</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E2:33_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <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">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="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">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="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">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 <b>and </b><br/><a NAME="E3:33_1"/><i><font color="Green" title="E30">A4</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c6">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E4:33_1"/><i><font color="Green" title="E31">A5</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E5:33_1"/><i><font color="Green" title="E32">A6</font></i>: 
( <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">a'</font>,<font color="Maroon" title="c11">b'</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">a'</font>,<font color="Maroon" title="c12">c'</font> )
 ;<br/>

<b>reconsider </b><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">a1'</font> = <font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c17">b1'</font> = <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c18">c1'</font> = <font color="Maroon" title="c12">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font> <b>by </b><i><a class="ref" href="papdesaf.html#T2" target="_self" title="PAPDESAF:th.2">Th2</a></i>;<br/>
<b>reconsider </b><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">u'</font> = <font color="Maroon" title="c16">a1'</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="papdesaf.html#E1:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self" title="PAPDESAF:th.4">Th4</a></i>;<br/>
<div><i><font color="Green" title="E33">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a'</font>
 ;<br/><a NAME="E2:33_1_1"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b'</font>
 ;<br/>

<a NAME="E2:33_1_1_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="c8">b</font>,<font color="Maroon" title="c11">b'</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E5:33_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="papdesaf.html#E1:33_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:33_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E4:33_1_1_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E3:33_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="papdesaf.html#E1:33_1_1_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:33_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E4:33_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:33_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1_2"/><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">c'</font>
 ;<br/>

<a NAME="E2:33_1_1_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="c9">c</font>,<font color="Maroon" title="c12">c'</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E5:33_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="papdesaf.html#E1:33_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:33_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E4:33_1_1_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E3:33_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="papdesaf.html#E1:33_1_1_2"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:33_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E4:33_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:33_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">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:33_1_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a'</font>
 ;<br/><a NAME="E2:33_1_2"/><i><font color="Green" title="E35">A13</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c14">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_2_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c14">b1</font>
 ;<br/>

<a NAME="E2:33_1_2_1"/><b>then </b>
 <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">a'</font>,<font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="ref" href="papdesaf.html#T3" target="_self" title="PAPDESAF:th.3">Th3</a></i>;<br/>
<a NAME="E3:33_1_2_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E3:33_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="papdesaf.html#E1:33_1_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:33_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E4:33_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:33_1_2"/><i><font color="Green" title="E36">A14</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c15">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_2_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c15">c1</font>
 ;<br/>

<a NAME="E2:33_1_2_2"/><b>then </b>
 <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">a'</font>,<font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="ref" href="papdesaf.html#T3" target="_self" title="PAPDESAF:th.3">Th3</a></i>;<br/>
<a NAME="E3:33_1_2_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E3:33_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="papdesaf.html#E1:33_1_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:33_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E4:33_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:33_1_2"/><i><font color="Green" title="E37">A15</font></i>: 
( <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">a1'</font>,<font color="Maroon" title="c17">b1'</font> &amp; <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">a1'</font>,<font color="Maroon" title="c18">c1'</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E5:33_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/><a NAME="E5:33_1_2"/><i><font color="Green" title="E38">A16</font></i>: 
( <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c17">b1'</font> &amp; <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">c1</font>,<font color="Maroon" title="c18">c1'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_2_3"/>
( <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">c'</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:33_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="papdesaf.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E2:33_1_2_3"/>
( <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c17">b1'</font> &amp; <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">c1</font>,<font color="Maroon" title="c18">c1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c23">v''</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font>;<br/><b>set </b><font color="Maroon" title="c24">w''</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font>;<br/><b>reconsider </b><font color="Maroon" title="c25">b1''</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c26">c1''</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">OAS</font> <b>by </b><i><a class="txt" href="papdesaf.html#E1:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self" title="PAPDESAF:th.4">Th4</a></i>;<br/><a NAME="E7:33_1_2"/>
( <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u'</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">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> &amp; <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">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> &amp; <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u'</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">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> &amp; <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">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#T25" title="ANALOAF:th.25">ANALOAF:25</a></i>;<br/><a NAME="E8:33_1_2"/><b>then </b>
( <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u'</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">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> &amp; <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">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> &amp; <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">u'</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">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> &amp; <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">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></i>;<br/><a NAME="E9:33_1_2"/><b>then </b>
( <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c25">b1''</font> &amp; <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">c1</font>,<font color="Maroon" title="c26">c1''</font> &amp; <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">a1'</font>,<font color="Maroon" title="c25">b1''</font> &amp; <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">a1'</font>,<font color="Maroon" title="c26">c1''</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></i>;<br/><a NAME="E10:33_1_2"/><b>then </b><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c25">b1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b1'</font> &amp; <font color="Maroon" title="c26">c1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c1'</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:33_1_2"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="papdesaf.html#E3:33_1_2"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="papdesaf.html#E4:33_1_2"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="papdesaf.html#E5:33_1_2"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="pasch.html#T12" title="PASCH:th.12">PASCH:12</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>
<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">u'</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c22">u'</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c20">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></i>
<br/>.= 
<font color="Maroon" title="c21">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c20">v</font>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>
;<br/><a NAME="E12:33_1_2"/><b>then </b>
<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">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T24" title="ANALOAF:th.24">ANALOAF:24</a></i>;<br/><a NAME="E13:33_1_2"/><b>then </b>
<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">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">u'</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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c19">u</font>
 <b>by </b><i><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></i>;<br/><a NAME="E14:33_1_2"/><b>then </b>
<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">b1'</font>,<font color="Maroon" title="c18">c1'</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="papdesaf.html#E10:33_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E15:33_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">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:33_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">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E8:33_1"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font> <a href="papdesaf.html#NR4" title="PAPDESAF:NR.4">satisfies_des'</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></i>;<br/>


</div>
