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<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W2</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 ;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E29</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<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>assume </b><a NAME="E3:18"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 ;<br/>

<b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</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/>
<a NAME="E4:18"/><i><font color="Green">E32</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><font color="Maroon">W1</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> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="pencil_4.html#T1" target="_self">Th1</a>, , </i>;<br/>
<a NAME="E5:18"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</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>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/>
<a NAME="E6:18"/>
<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">W2</font>
 
<b>by </b><i><a class="ref" href="pencil_4.html#T1" target="_self">Th1</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:18"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:18"/><b>then </b>
 <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T20">VECTSP_9:20</a></i>;<br/>
<a NAME="E9:18"/><b>then </b>
<font color="Maroon">W1</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> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_5.html#T40">VECTSP_5:40</a></i>;<br/>
<a NAME="E10:18"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">W1</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="hidden.html#R2">in</a>  <a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:18"/>
( <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T31">VECTSP_9:31</a></i>;<br/>
<a NAME="E12:18"/><b>then </b>
( <font color="Maroon">W1</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E13:18"/><b>then </b>
not <font color="Maroon">W1</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="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W1</font></span>)</span>,<span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W2</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E14:18"/><b>then </b>
<font color="Maroon">W1</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="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W1</font></span>)</span>,<span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W2</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E15:18"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">W1</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="hidden.html#R2">in</a>  <a href="pencil_4.html#K2" target="_self">pencil</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
 
;<br/>
<a NAME="E16:18"/>
<font color="Maroon">W1</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="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/>
<a NAME="E17:18"/><b>then </b>
<font color="Maroon">W1</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="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pencil_4.html#K2" target="_self">pencil</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_4.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E18:18"/>
<font color="Maroon">W1</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="hidden.html#R2">in</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>,<font color="Maroon">k</font>
 ;<br/>


</div>
