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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">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<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>assume </b><a NAME="E1:8"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">W1</font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3">+</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:8"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E5:8"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T79">RLSUB_1:79</a></i>;<br/>
<a NAME="E6:8"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , <a class="ref" href="rlsub_1.html#T16">RLSUB_1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3">+</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T79">RLSUB_1:79</a></i>;<br/>


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