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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>









<a NAME="E1:22"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_4.html#T12">RLVECT_4:12</a></i>;<br/>
<b>assume </b><a NAME="E2:22"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:22"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="rlsub_2.html#R1">is_the_direct_sum_of</a> <font color="Maroon">c<sub>5</sub></font>, <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="hahnban.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E4:22"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="rlsub_2.html#K4">|--</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p4">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="hahnban.html#E1:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hahnban.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E5:22"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlsub_2.html#K4">|--</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T19">RLSUB_1:19</a></i>;<br/>


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