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<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>











<b>assume </b><a NAME="E1:32"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp; not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a> <font color="Maroon">W</font>, <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/>





<b>assume </b><a NAME="E3:32"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">w</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</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"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="vectsp_4.html#T18">VECTSP_4:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18">VECTSP_4:18</a></i>;<br/>
<i><font color="Green">E46</font></i>: <font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font> = 
<font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="vectsp_4.html#T22">VECTSP_4:22</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="vectsp_4.html#T22">VECTSP_4:22</a></i>
;<br/>
<a NAME="E7:32"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T22">VECTSP_4:22</a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</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="E9:32"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:32"/>
<font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T22">VECTSP_4:22</a></i>;<br/>
<b>then </b><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font> = 
<font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/>
<b>hence </b><a NAME="E12:32"/>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</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"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , , , , </i>;<br/>


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