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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">W1</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>thus </b><a NAME="E1:3"/>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font> implies  ex <font color="Olive">v1</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">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">v1</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><br/><a NAME="E5:3_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 ;<br/>
<b>take </b>
<font color="Maroon">v1</font>
;<br/>

<b>take </b>
<font color="Maroon">v2</font>
;<br/>

<b>thus </b><a NAME="E6:3_1"/>
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> )
 <b>by </b><i/>;<br/>


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

<b>given </b><font color="Maroon">v1</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="E2:3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> )
 ;<br/>

<a NAME="E3:3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:3"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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