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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">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>assume </b><a NAME="E1:49"/><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/>

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





<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><b>take </b><font color="Maroon">u1</font> = <font color="Maroon">w</font> <a href="analort.html#K1" target="_self">+</a> <font color="Maroon">x</font>;<br/><a NAME="E2:49_1"/>
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u1</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#T18">ANALOAF:18</a></i>;<br/><a NAME="E3:49_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">w</font>,<font color="Maroon">u1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font>
 ;<br/><a NAME="E2:49_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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#T22">RLVECT_1:22</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:49_1"/>
 ex <font color="Olive">u1</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Maroon">w</font>,<font color="Olive">u1</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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E46</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="E3:49_2"/><i><font color="Green">E47</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="E5:49_2"/><i><font color="Green">E48</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/><b>take </b><font color="Maroon">u1</font> = <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="analort.html#K1" target="_self">+</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>;<br/><a NAME="E6:49_2"/>
<font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">w</font>,<font color="Maroon">u1</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T25">ANALOAF:25</a></i>;<br/><a NAME="E7:49_2"/><b>then </b>
<font color="Maroon">w</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T21">ANALOAF:21</a></i>;<br/><a NAME="E8:49_2"/><b>then </b>
 <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u1</font> <a href="analoaf.html#R1">//</a>  <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>
 <b>by </b><i>, </i>;<br/><a NAME="E9:49_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">w</font>,<font color="Maroon">u1</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>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font>
 ;<br/><a NAME="E2:49_2_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E3:49_2_1"/><b>then </b>
<font color="Maroon">v2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</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#T22">RLVECT_1:22</a></i>;<br/><b>hence </b><a NAME="E4:49_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:49_2"/>
 ex <font color="Olive">u1</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Maroon">w</font>,<font color="Olive">u1</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
 ex <font color="Olive">u1</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Maroon">w</font>,<font color="Olive">u1</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/>


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