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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<b>given </b><font color="Maroon">p</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">q</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:17"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</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">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 ;<br/><a NAME="E2:17_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Maroon">u</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:17_1_1"/><b>then </b>
( ( <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> or <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> ) &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:17_1_1"/>
 ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Olive">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Olive">y</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1"/>
 ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Olive">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Olive">y</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>;<br/><a NAME="E2:17_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:17_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E3:17_2_1"/><b>then </b>
<font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</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="E4:17_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="E4:17_2"/>
 ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Olive">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Olive">y</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17"/>
 ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Olive">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>,<font color="Olive">y</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> )
 <b>by </b><i/>;<br/>


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