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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">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v2</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="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/>







<b>thus </b><a NAME="E1:69"/>
(  ex <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Olive">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Olive">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font> ) implies <font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:69_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E2:69_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>

<b>consider </b><font color="Maroon">u1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:69_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E5:69_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
 <b>by </b><i>, </i>;<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="E7:69_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E8:69_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font>
 <b>by </b><i>, </i>;<br/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_1.html#T17" target="_self">Th17</a>, <a class="ref" href="rlsub_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T44">RLVECT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E10:69_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T18" target="_self">Th18</a>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:69"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:69"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">v1</font>
;<br/>

<b>thus </b><a NAME="E4:69"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>

<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font></span>)</span></span>)</span> = 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E6:69"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


</div>
