<?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">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>











<b>assume </b><a NAME="E1:32"/><i><font color="Green">E45</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

<b>set </b><font color="Maroon">a</font> = <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:32_1"/>
( <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font>,<font color="Maroon">u1</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/><a NAME="E3:32_1"/><b>then </b>
( <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u1</font>,<font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:32_1"/><b>then </b>
<font color="Maroon">u</font>,<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:32_1"/>
 ex <font color="Olive">t</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Olive">t</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">t</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:32_2"/>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1"/>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 ;<br/>

<a NAME="E2:32_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>
 
;<br/>
<b>hence </b><a NAME="E3:32_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_2"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:32_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span> <a href="analmetr.html#R2">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T17">ANALMETR:17</a></i>;<br/><b>set </b><font color="Maroon">b</font> = <font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>;<br/><a NAME="E6:32_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 <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> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:32_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </b></a><div class="add">

<font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<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"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<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"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E2:32_2_2"/>
<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E8:32_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>;<br/><a NAME="E9:32_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>;<br/><a NAME="E10:32_2"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span> <a href="analmetr.html#R2">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>;<br/><a NAME="E11:32_2"/>
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font> <a href="analmetr.html#R2">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="analmetr.html#T11">ANALMETR:11</a></i>;<br/><a NAME="E12:32_2"/><b>then </b>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D3">ANALMETR:def 3</a></i>;<br/><a NAME="E13:32_2"/><b>then </b>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E14:32_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">r</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> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>then </b><i><font color="Green">E96</font></i>: <font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> = 
2 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></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> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E17:32_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</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">v</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_5"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:32_2_5"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> &amp; <span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/>

<b>hence </b><span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</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"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</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"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E3:32_2_5"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_2_5_2"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>
 <b>by </b><i>, <a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>

<a NAME="E2:32_2_5_2"/>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:32_2_5_2"/>
<font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E4:32_2_5"/><b>then </b>
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="geomtrap.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E5:32_2_5"/>
 ex <font color="Olive">t</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Olive">t</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">t</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_6"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:32_2_6"/>
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="analmetr.html#R2">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="geomtrap.html#D1" target="_self">Def1</a>, <a class="ref" href="geomtrap.html#T2" target="_self">Th2</a>, <a class="ref" href="analmetr.html#T11">ANALMETR:11</a></i>;<br/><a NAME="E3:32_2_6"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D3">ANALMETR:def 3</a></i>;<br/><a NAME="E4:32_2_6"/><b>then </b>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:32_2_6"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:32_2_6"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E7:32_2_6"/>
<font color="Maroon">r</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">v</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E8:32_2_6"/><b>then </b>
( <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font> or <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T18">ANALMETR:18</a></i>;<br/><a NAME="E9:32_2_6"/><b>then </b>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#T21">ANALOAF:21</a></i>;<br/><b>hence </b><a NAME="E10:32_2_6"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font>,<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#T1" target="_self">Th1</a>, , <a class="txt" href="geomtrap.html#E1:32"><i><font color="Green">E45</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E20:32_2"/>
 ex <font color="Olive">t</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Olive">t</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">t</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32"/>
 ex <font color="Olive">t</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Olive">t</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">t</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>


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