<?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="E1:29"/><i><font color="Green" title="E23">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="E2:29"/>
for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</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="b3">o</font>, <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">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">a'</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">b'</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">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">M</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</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">N</font> &amp; <font color="Olive" title="b9">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b7">a'</font> &amp; <font color="Olive" title="b5">b</font>,<font color="Olive" title="b9">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b8">b'</font> holds <br/><font color="Olive" title="b4">a</font>,<font color="Olive" title="b9">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b7">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">M</font>, <font color="Maroon" title="c5">N</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="c6">o</font>, <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="E1:29_1"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c4">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E2:29_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">N</font>
 <b>and </b><br/><a NAME="E3:29_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> )
 <b>and </b><br/><a NAME="E4:29_1"/><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a'</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">c'</font> )
 <b>and </b><br/><a NAME="E5:29_1"/><i><font color="Green" title="E28">A6</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">M</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</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">N</font> &amp; <font color="Maroon" title="c12">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> )
 <b>and </b><br/><a NAME="E6:29_1"/><i><font color="Green" title="E29">A7</font></i>: 
( <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c11">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">a'</font> &amp; <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c11">b'</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">o1</font> = <font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c16">c1</font> = <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c17">a1'</font> = <font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c18">b1'</font> = <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c19">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="c20">q</font> = <font color="Maroon" title="c13">o1</font>, <font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c14">a1</font>, <font color="Maroon" title="c22">v</font> = <font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c23">w</font> = <font color="Maroon" title="c16">c1</font>, <font color="Maroon" title="c24">u'</font> = <font color="Maroon" title="c17">a1'</font>, <font color="Maroon" title="c25">v'</font> = <font color="Maroon" title="c18">b1'</font>, <font color="Maroon" title="c26">w'</font> = <font color="Maroon" title="c19">c1'</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#E1:29"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self" title="PAPDESAF:th.4">Th4</a></i>;<br/>
<a NAME="E9:29_1"/>
( <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c18">b1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">a1'</font> &amp; <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c19">c1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c18">b1'</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:29_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>
<a NAME="E10:29_1"/><b>then </b>
( <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c18">b1'</font> &amp; <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c19">c1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c18">b1'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E11:29_1"/><b>then </b><i><font color="Green" title="E30">A8</font></i>: 
( <font color="Maroon" title="c22">v</font>,<font color="Maroon" title="c24">u'</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c25">v'</font> &amp; <font color="Maroon" title="c22">v</font>,<font color="Maroon" title="c26">w'</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c23">w</font>,<font color="Maroon" title="c25">v'</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:29"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></i>;<br/>
<a NAME="E12:29_1"/><i><font color="Green" title="E31">A9</font></i>: 
( <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c21">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> &amp; <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c23">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c8">b</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:29_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="papdesaf.html#E3:29_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="papdesaf.html#E5:29_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E2:29_1_1"/><b>then </b>
( <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
( <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c16">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1"/>
( <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c21">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> &amp; <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c23">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:29"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c27">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E14:29_1"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c21">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">r1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span> &amp; <font color="Maroon" title="c27">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:29_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="papdesaf.html#E36"><i><font color="Green" title="E20">Lm2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E16:29_1"/><i><font color="Green" title="E33">A11</font></i>: 
( <font color="Maroon" title="c23">w</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">r2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span> &amp; <font color="Maroon" title="c28">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:29_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="papdesaf.html#E12:29_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="papdesaf.html#E36"><i><font color="Green" title="E20">Lm2</font></i></a></i>;<br/>
<a NAME="E17:29_1"/><i><font color="Green" title="E34">A12</font></i>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c27">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="papdesaf.html#E14:29_1"><i><font color="Green" title="E32">A10</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c27">r1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c22">v</font></span>)</span> = 
<font color="Maroon" title="c27">r1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c22">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38" title="RLVECT_1:th.38">RLVECT_1:38</a></i>
<br/>.= 
<font color="Maroon" title="c21">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font>
<b>by </b><i><a class="txt" href="papdesaf.html#E14:29_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
;<br/>
<a NAME="E19:29_1"/><b>then </b><i><font color="Green" title="E35">A13</font></i>: 
<font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c27">r1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E17:29_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="analoaf.html#T13" title="ANALOAF:th.13">ANALOAF:13</a></i>;<br/>
<a NAME="E20:29_1"/><i><font color="Green" title="E36">A14</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c27">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c28">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E14:29_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="papdesaf.html#E16:29_1"><i><font color="Green" title="E33">A11</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c28">r2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c22">v</font></span>)</span> = 
<font color="Maroon" title="c28">r2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c22">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38" title="RLVECT_1:th.38">RLVECT_1:38</a></i>
<br/>.= 
<font color="Maroon" title="c23">w</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font>
<b>by </b><i><a class="txt" href="papdesaf.html#E16:29_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
;<br/>
<a NAME="E22:29_1"/><b>then </b><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c28">r2</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">w</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E20:29_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="analoaf.html#T12" title="ANALOAF:th.12">ANALOAF:12</a></i>;<br/>
<a NAME="E23:29_1"/><i><font color="Green" title="E38">A16</font></i>: 
( <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c27">r1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c28">r2</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E20:29_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="xcmplx_1.html#T203" title="XCMPLX_1:th.203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E24:29_1"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c27">r1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E14:29_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="papdesaf.html#E16:29_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="xcmplx_1.html#T203" title="XCMPLX_1:th.203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E25:29_1"/><i><font color="Green" title="E40">A18</font></i>: 
( <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c24">u'</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c25">v'</font> &amp; <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c26">w'</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c25">v'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_4"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c11">b'</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:29_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="papdesaf.html#E3:29_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="papdesaf.html#E5:29_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E2:29_1_4"/><b>then </b>
( <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c10">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c12">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c11">b'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:29_1_4"/><b>then </b>
( <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c18">b1'</font> &amp; <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c19">c1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c18">b1'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_4"/>
( <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c24">u'</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c25">v'</font> &amp; <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c26">w'</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c25">v'</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:29"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:29_1"/><i><font color="Green" title="E41">A19</font></i>: 
( not <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c26">w'</font> &amp; not <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c24">u'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_5"/>
( <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c26">w'</font> or <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c22">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c20">q</font>,<font color="Maroon" title="c24">u'</font> )
 ;<br/>

<a NAME="E2:29_1_5"/><b>then </b>
( <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c19">c1'</font> or <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:29"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="papdesaf.html#T5" target="_self" title="PAPDESAF:th.5">Th5</a></i>;<br/>
<a NAME="E3:29_1_5"/><b>then </b>
( <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c12">c'</font> or <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c10">a'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>
<a NAME="E4:29_1_5"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">c'</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">a'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:29_1_5"/><b>then </b>
( <font color="Maroon" title="c12">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> or <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:29_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="papdesaf.html#E3:29_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="papdesaf.html#E4:29_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="papdesaf.html#E5:29_1"><i><font color="Green" title="E28">A6</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="E6:29_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:29_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="papdesaf.html#E2:29_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="papdesaf.html#E3:29_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="papdesaf.html#E4:29_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="papdesaf.html#E5:29_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c29">s</font> = <font color="Maroon" title="c27">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>;<br/>
<font color="Maroon" title="c25">v'</font> = 
<font color="Maroon" title="c20">q</font> <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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c28">r2</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E11:29_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="papdesaf.html#E22:29_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="papdesaf.html#E23:29_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="papdesaf.html#E25:29_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="papdesaf.html#E26:29_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="papdesaf.html#T15" target="_self" title="PAPDESAF:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c20">q</font> <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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224" title="XCMPLX_1:th.224">XCMPLX_1:224</a></i>
<br/>.= 
<font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>

;<br/>
<a NAME="E28:29_1"/><b>then </b><i><font color="Green" title="E42">A20</font></i>: 
<font color="Maroon" title="c25">v'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">r2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</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/>
<font color="Maroon" title="c25">v'</font> = 
<font color="Maroon" title="c20">q</font> <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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c27">r1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E11:29_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="papdesaf.html#E19:29_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="papdesaf.html#E23:29_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="papdesaf.html#E25:29_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="papdesaf.html#E26:29_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="papdesaf.html#T15" target="_self" title="PAPDESAF:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c20">q</font> <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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224" title="XCMPLX_1:th.224">XCMPLX_1:224</a></i>
<br/>.= 
<font color="Maroon" title="c20">q</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>

;<br/>
<a NAME="E30:29_1"/><b>then </b>
<font color="Maroon" title="c25">v'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">r1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</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/>
<b>then </b><i><font color="Green" title="E43">A21</font></i>: <font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E14:29_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="papdesaf.html#E28:29_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="analoaf.html#T13" title="ANALOAF:th.13">ANALOAF:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c28">r2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E32:29_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c28">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="papdesaf.html#E16:29_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="papdesaf.html#E24:29_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E44">A22</font></i>: <font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c28">r2</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E31:29_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="analoaf.html#T13" title="ANALOAF:th.13">ANALOAF:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T205" title="XCMPLX_1:th.205">XCMPLX_1:205</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span>

;<br/>
<i><font color="Green" title="E45">A23</font></i>: <font color="Maroon" title="c21">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font> = 
<font color="Maroon" title="c27">r1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">w</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E14:29_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="papdesaf.html#E16:29_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="analoaf.html#T13" title="ANALOAF:th.13">ANALOAF:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">w</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/>
1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c21">u</font></span>)</span> = 
<font color="Maroon" title="c26">w'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c21">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">w</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E33:29_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="papdesaf.html#E34:29_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="papdesaf.html#E37"><i><font color="Green" title="E21">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">w</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c20">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T48" title="RLVECT_1:th.48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">r2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">u'</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c23">w</font></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E37"><i><font color="Green" title="E21">Lm3</font></i></a></i>
;<br/>
<a NAME="E36:29_1"/><b>then </b>
( <font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c26">w'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c23">w</font>,<font color="Maroon" title="c24">u'</font> or <font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c26">w'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c24">u'</font>,<font color="Maroon" title="c23">w</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T18" title="ANALMETR:th.18">ANALMETR:18</a></i>;<br/>
<a NAME="E37:29_1"/><b>then </b>
<font color="Maroon" title="c21">u</font>,<font color="Maroon" title="c26">w'</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c23">w</font>,<font color="Maroon" title="c24">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="E38:29_1"/><b>then </b>
<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c19">c1'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c17">a1'</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:29"><i><font color="Green" title="E23">A1</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="E39:29_1"/>
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c12">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">a'</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="E3:29"/>
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font> <a href="papdesaf.html#NR1" title="PAPDESAF:NR.1">satisfies_PAP'</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D2" title="AFF_2:def.2">AFF_2:def 2</a></i>;<br/>


</div>
