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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E1:17"/>
<font color="Maroon" title="c1">V</font> <a href="rlsub_2.html#R1" title="RLSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">V</font> <a href="rlsub_2.html#R1" title="RLSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W2</font>,<font color="Maroon" title="c2">W1</font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T46" title="RLSUB_2:th.46">RLSUB_2:46</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E3:17"/>
<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 ;<br/>

<a NAME="E4:17"/><b>then </b>
<font color="Maroon" title="c4">v</font> <a href="rlsub_2.html#K4" title="RLSUB_2:func.4">|--</a> <font color="Maroon" title="c3">W2</font>,<font color="Maroon" title="c2">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">v</font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="hahnban.html#E2:17"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="hahnban.html#T19" target="_self" title="HAHNBAN:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon" title="c4">v</font> <a href="rlsub_2.html#K4" title="RLSUB_2:func.4">|--</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="hahnban.html#E2:17"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="hahnban.html#T18" target="_self" title="HAHNBAN:th.18">Th18</a></i>;<br/>


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