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<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">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">u2</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="E1:36"/><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="E2:36"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:36"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u2</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Maroon">v</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">u2</font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font>,<font color="Maroon">v</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">u2</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>set </b><font color="Maroon">b</font> = <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>;<br/>
<b>set </b><font color="Maroon">c</font> = <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font>;<br/>
<b>assume </b><a NAME="E4:36"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u2</font>
 ;<br/>

<a NAME="E5:36"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:36"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u2</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> )
 
<b>by </b><i>, <a class="ref" href="analoaf.html#T21">ANALOAF:21</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:36"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u2</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T17">ANALOAF:17</a>, <a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>
<a NAME="E9:36"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font>
 ;<br/><a NAME="E2:36_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u2</font>
 ;<br/><b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</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>, <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_1"/>
<font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:36"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>
 ;<br/><a NAME="E2:36_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u1</font>
 ;<br/><b>hence </b><a NAME="E3:36_2_1"/>
<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>
 <b>by </b><i>, <a class="ref" href="geomtrap.html#T2" target="_self">Th2</a>, <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_2"/>
<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:36"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E12:36"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">u1</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">u2</font> &amp; <font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:36"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:36"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u1</font>
 ;<br/><a NAME="E2:36_3"/><b>then </b>
<font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</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">u2</font>
 <b>by </b><i>, , <a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E45</font></i></a>, <a class="txt" href="geomtrap.html#E2:36"><i><font color="Green">E46</font></i></a>, </i>;<br/><a NAME="E3:36_3"/><b>then </b>
<font color="Maroon">v</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">u2</font>,<font color="Maroon">v</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">u2</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="E4:36_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="geomtrap.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u2</font>
 ;<br/><a NAME="E2:36_4"/><b>then </b>
<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">v</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">u2</font>
 <b>by </b><i>, , <a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E45</font></i></a>, <a class="txt" href="geomtrap.html#E3:36"><i><font color="Green">E47</font></i></a>, </i>;<br/><a NAME="E3:36_4"/><b>then </b>
<font color="Maroon">v</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">u2</font>,<font color="Maroon">v</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">u2</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="E4:36_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="geomtrap.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:36"/>
contradiction
 <b>by </b><i>, <a class="ref" href="geomtrap.html#T3" target="_self">Th3</a></i>;<br/>


</div>
