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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">W</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:38"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><br/> for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="rusub_4.html#K3" target="_self">Up</a> <font color="Maroon">W</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="rusub_4.html#K3" target="_self">Up</a> <font color="Maroon">W</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="rusub_4.html#K3" target="_self">Up</a> <font color="Maroon">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:38_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rusub_4.html#K3" target="_self">Up</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="rusub_4.html#K3" target="_self">Up</a> <font color="Maroon">W</font> )
 ;<br/>

<a NAME="E2:38_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 <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> )
 
;<br/>
<a NAME="E3:38_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:38_1"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</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#T29">RLSUB_1:29</a></i>;<br/>
<a NAME="E5:38_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T28">RLSUB_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E7:38_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:38_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="rusub_4.html#K3" target="_self">Up</a> <font color="Maroon">W</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:38"/>
 <a href="rusub_4.html#K3" target="_self">Up</a> <font color="Maroon">W</font> is <a href="rusub_4.html#V2" target="_self">Affine</a>
 <b>by </b><i/>;<br/>

<a NAME="E3:38"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</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#T25">RLSUB_1:25</a></i>;<br/>
<b>hence </b><a NAME="E4:38"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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