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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">OAS</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">OAS</font> <a href="hidden.html#R1">=</a>  <a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font>
 ;<br/>

<b>set </b><font color="Maroon">AS</font> =  <a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font>;<br/>
<a NAME="E2:29"/>
for <font color="Olive">M</font>, <font color="Olive">N</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span><br/> for <font color="Olive">o</font>, <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">a'</font>, <font color="Olive">b'</font>, <font color="Olive">c'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>  st <font color="Olive">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">N</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">M</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a'</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b'</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c'</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">M</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">M</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">M</font> &amp; <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> &amp; <font color="Olive">c'</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> &amp; <font color="Olive">a</font>,<font color="Olive">b'</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">a'</font> &amp; <font color="Olive">b</font>,<font color="Olive">c'</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">b'</font> holds <br/><font color="Olive">a</font>,<font color="Olive">c'</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">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">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>;<br/><b>let </b><font color="Maroon">o</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">a'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">b'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">c'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E2:29_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E3:29_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E4:29_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> )
 <b>and </b><br/><a NAME="E5:29_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E6:29_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b'</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o</font>, <font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">c1</font> = <font color="Maroon">c</font>, <font color="Maroon">a1'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b'</font>, <font color="Maroon">c1'</font> = <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">o1</font>, <font color="Maroon">u</font> = <font color="Maroon">a1</font>, <font color="Maroon">v</font> = <font color="Maroon">b1</font>, <font color="Maroon">w</font> = <font color="Maroon">c1</font>, <font color="Maroon">u'</font> = <font color="Maroon">a1'</font>, <font color="Maroon">v'</font> = <font color="Maroon">b1'</font>, <font color="Maroon">w'</font> = <font color="Maroon">c1'</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<a NAME="E9:29_1"/>
( <font color="Maroon">a1</font>,<font color="Maroon">b1'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b1</font>,<font color="Maroon">a1'</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">c1'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c1</font>,<font color="Maroon">b1'</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:29_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>
<a NAME="E10:29_1"/><b>then </b>
( <font color="Maroon">b1</font>,<font color="Maroon">a1'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a1</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">c1'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c1</font>,<font color="Maroon">b1'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E11:29_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u'</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">u</font>,<font color="Maroon">v'</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">w'</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">w</font>,<font color="Maroon">v'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:29_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">w</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">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">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="txt" href="papdesaf.html#E1:29"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E2:29_1_1"/><b>then </b>
( <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
( <font color="Maroon">o1</font>,<font color="Maroon">a1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">o1</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">o1</font>,<font color="Maroon">c1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">o1</font>,<font color="Maroon">b1</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1"/>
( <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">w</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">v</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:29_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E16:29_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E17:29_1"/><i><font color="Green">E68</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="papdesaf.html#T5" target="_self">Th5</a></i>;<br/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> = 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38">RLVECT_1:38</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="papdesaf.html#T5" target="_self">Th5</a>, <a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
;<br/>
<a NAME="E19:29_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="analoaf.html#T13">ANALOAF:13</a></i>;<br/>
<a NAME="E20:29_1"/><i><font color="Green">E70</font></i>: 
(  <a href="real_1.html#K1">-</a> <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="real_1.html#K1">-</a> <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="papdesaf.html#T5" target="_self">Th5</a>, </i>;<br/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> = 
<font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38">RLVECT_1:38</a></i>
<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
;<br/>
<a NAME="E22:29_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="analoaf.html#T12">ANALOAF:12</a></i>;<br/>
<a NAME="E23:29_1"/><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E24:29_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">r1</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">r2</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="papdesaf.html#T5" target="_self">Th5</a>, , <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E25:29_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">q</font>,<font color="Maroon">u'</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">v'</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">w'</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">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">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , <a class="txt" href="papdesaf.html#E1:29"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E2:29_1_4"/><b>then </b>
( <font color="Maroon">o</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:29_1_4"/><b>then </b>
( <font color="Maroon">o1</font>,<font color="Maroon">a1'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">o1</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">o1</font>,<font color="Maroon">c1'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">o1</font>,<font color="Maroon">b1'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_4"/>
( <font color="Maroon">q</font>,<font color="Maroon">u'</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">v'</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">w'</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">v'</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:29_1"/><i><font color="Green">E75</font></i>: 
( not <font color="Maroon">q</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">w'</font> &amp; not <font color="Maroon">q</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">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">q</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">w'</font> or <font color="Maroon">q</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">u'</font> )
 ;<br/>

<a NAME="E2:29_1_5"/><b>then </b>
( <font color="Maroon">o1</font>,<font color="Maroon">b1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">o1</font>,<font color="Maroon">c1'</font> or <font color="Maroon">o1</font>,<font color="Maroon">b1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">o1</font>,<font color="Maroon">a1'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:29_1_5"/><b>then </b>
( <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c'</font> or <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>
<a NAME="E4:29_1_5"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">c'</font> or  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">a'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:29_1_5"/><b>then </b>
( <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> or <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, , , <a class="txt" href="papdesaf.html#E1:29"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T39">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"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">s</font> = <font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span>;<br/>
<font color="Maroon">v'</font> = 
<font color="Maroon">q</font> <a href="rlvect_1.html#K4">+</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">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="papdesaf.html#T4" target="_self">Th4</a>, , <a class="txt" href="papdesaf.html#E2:29_1"><i><font color="Green">E38</font></i></a>, , , </i>
<br/>.= 
<font color="Maroon">q</font> <a href="rlvect_1.html#K4">+</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">-</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224">XCMPLX_1:224</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>

;<br/>
<a NAME="E28:29_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">v'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>;<br/>
<font color="Maroon">v'</font> = 
<font color="Maroon">q</font> <a href="rlvect_1.html#K4">+</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">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="papdesaf.html#T4" target="_self">Th4</a>, , <a class="txt" href="papdesaf.html#E2:29_1"><i><font color="Green">E38</font></i></a>, , , </i>
<br/>.= 
<font color="Maroon">q</font> <a href="rlvect_1.html#K4">+</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">-</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224">XCMPLX_1:224</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>

;<br/>
<a NAME="E30:29_1"/><b>then </b>
<font color="Maroon">v'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>;<br/>
<b>then </b><i><font color="Green">E77</font></i>: <font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="papdesaf.html#T5" target="_self">Th5</a>, , <a class="ref" href="analoaf.html#T13">ANALOAF:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E32:29_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>then </b><i><font color="Green">E78</font></i>: <font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E3:29_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="analoaf.html#T13">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">r1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T205">XCMPLX_1:205</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>

;<br/>
<i><font color="Green">E79</font></i>: <font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font> = 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="papdesaf.html#T5" target="_self">Th5</a>, , <a class="ref" href="analoaf.html#T13">ANALOAF:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">w'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="papdesaf.html#D5" target="_self">Def5</a>, <a class="ref" href="papdesaf.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="papdesaf.html#T3" target="_self">Th3</a></i>
;<br/>
<a NAME="E36:29_1"/><b>then </b>
( <font color="Maroon">u</font>,<font color="Maroon">w'</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">w</font>,<font color="Maroon">u'</font> or <font color="Maroon">u</font>,<font color="Maroon">w'</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u'</font>,<font color="Maroon">w</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T18">ANALMETR:18</a></i>;<br/>
<a NAME="E37:29_1"/><b>then </b>
<font color="Maroon">u</font>,<font color="Maroon">w'</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">w</font>,<font color="Maroon">u'</font>
 
<b>by </b><i><a class="ref" href="geomtrap.html#D1">GEOMTRAP:def 1</a></i>;<br/>
<a NAME="E38:29_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">c1'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c1</font>,<font color="Maroon">a1'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E39:29_1"/>
<font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
 <a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font> <a href="papdesaf.html#NR1" target="_self">satisfies_PAP'</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D2">AFF_2:def 2</a></i>;<br/>


</div>
