<?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">x</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><a NAME="E1:38"/><i><font color="Green">E39</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:38"/>
( <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> iff  ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<a NAME="E1:38_1"/><i><font color="Green">E40</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> implies  ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1"/><i><font color="Green">E44</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>
 ;<br/>

<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_1"/><i><font color="Green">E46</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#R1">=</a> <font color="Maroon">v1</font> )
 ;<br/><b>take </b><font color="Maroon">u2</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">x</font>;<br/><a NAME="E2:38_1_1_1"/>
(  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp;  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> )
 <b>by </b><i>, <a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/><a NAME="E3:38_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E4:38_1_1_1"/>
<font color="Maroon">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:38_1_1_1"/>
 ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><b>consider </b><font color="Maroon">u2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:38_1_1_2"/><i><font color="Green">E79</font></i>: 
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">v2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:38_1_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:38_1_1_2"/>
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</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="analoaf.html#T17">ANALOAF:17</a></i>;<br/><a NAME="E7:38_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E8:38_1_1_2"/>
<font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E9:38_1_1_2"/>
 ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="txt" href="analort.html#E1"><i><font color="Green">Lemma38</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font>
 ;<br/><b>consider </b><font color="Maroon">u2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:38_1_1_3"/><i><font color="Green">E83</font></i>: 
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">v2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:38_1_1_3"/><i><font color="Green">E84</font></i>: 
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:38_1_1_3"/>
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font>
 <b>by </b><i>, , <a class="ref" href="analoaf.html#T17">ANALOAF:17</a></i>;<br/><a NAME="E7:38_1_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E8:38_1_1_3"/>
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</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="analoaf.html#T21">ANALOAF:21</a></i>;<br/><a NAME="E9:38_1_1_3"/><b>then </b>
<font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:38_1_1_3"/>
 ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="txt" href="analort.html#E1"><i><font color="Green">Lemma38</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:38_1_1"/>
 ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:38_1"/>
(  ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) implies <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">u2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:38_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u2</font>,<font color="Maroon">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:38_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:38_1_2"/>
(  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp;  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u2</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:38_1_2"/>
<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>
 <b>by </b><i>, <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_1"/>
( <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> iff  ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
( <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> iff  ex <font color="Olive">u2</font>, <font color="Olive">v2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Olive">u2</font>,<font color="Olive">v2</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) )
 ;<br/>


</div>
