<?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>let </b><font color="Maroon">v1</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="E1:39"/><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="E2:39"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</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">p</font> = <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>;<br/>
<b>set </b><font color="Maroon">q</font> = <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>;<br/>
<b>set </b><font color="Maroon">r</font> = <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>;<br/>
<b>set </b><font color="Maroon">s</font> = <font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>;<br/>
<a NAME="E3:39"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> &amp; <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> <font color="Maroon">v1</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">v1</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> <font color="Maroon">v1</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="geomtrap.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E4:39"/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R1">=</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="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:39"/><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="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:39"/><b>then </b><i><font color="Green">E67</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="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:39"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="geomtrap.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E8:39"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:39"/>
<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="geomtrap.html#T1" target="_self">Th1</a>, <a class="ref" href="analoaf.html#T21">ANALOAF:21</a></i>;<br/>
<a NAME="E10:39"/><b>then </b>
( <font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font> &amp; <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u</font> &amp; <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E11:39"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_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">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>
 ;<br/><b>then </b><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v</font> = 
<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="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:39_1"/>
( <font color="Maroon">u1</font>,<font color="Maroon">v1</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"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span> <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">v</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"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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="E13:39"/><i><font color="Green">E90</font></i>: 
( <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">u1</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font> implies ( <font color="Maroon">u1</font>,<font color="Maroon">v1</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"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span> <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">v</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"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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>, <a class="ref" href="geomtrap.html#T1" target="_self">Th1</a>, <a class="ref" href="geomtrap.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E14:39"/><b>then </b><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">v</font> implies <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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="E15:39"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> implies <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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>, , , <a class="txt" href="geomtrap.html#E1:39"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<a NAME="E16:39"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> implies <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="geomtrap.html#K1" target="_self">#</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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/>
<b>hence </b><a NAME="E17:39"/>
<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">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</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="txt" href="geomtrap.html#E1:39"><i><font color="Green">E45</font></i></a>, <a class="txt" href="geomtrap.html#E2:39"><i><font color="Green">E46</font></i></a>, <a class="txt" href="geomtrap.html#E3:39"><i><font color="Green">E47</font></i></a>, </i>;<br/>


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