<?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">v1</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">v2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E45</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">v1</font>,<font color="Maroon">u</font>,<font color="Maroon">u</font>,<font color="Maroon">v2</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

<b>set </b><font color="Maroon">a</font> = <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font>;<br/>
<a NAME="E3:30"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">v1</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font>,<font color="Maroon">u</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</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>,<font color="Maroon">v2</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</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="E4:30"/>
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> implies ( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:30_1"/>
<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:30_1"/><b>then </b>
( <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:30_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:30_1"/>
( <font color="Maroon">u</font>,<font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:30_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">v1</font>,<font color="Maroon">u</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v2</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:30_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font>
 ;<br/>

<a NAME="E2:30_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>
 ;<br/>

<a NAME="E2:30_1_1_1"/><b>then </b>
<font color="Maroon">v1</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">u</font>
 
;<br/>
<b>hence </b><a NAME="E3:30_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_1"/>
<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</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="E4:30_1_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <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>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E5:30_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:30_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 ;<br/>

<a NAME="E2:30_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2_1"/>
<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">v2</font>
 ;<br/>

<a NAME="E2:30_1_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>
 
;<br/>
<b>hence </b><a NAME="E3:30_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_2"/>
<font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</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="E4:30_1_2"/><b>then </b>
<font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v2</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="E5:30_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:30_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 <b>by </b><i>, </i>;<br/>


</div>
