<?xml version="1.0"?>
<div class="add">

<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>assume </b><a NAME="E1:26"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</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="E2:26"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E3:26"/><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> <a href="hidden.html#R1">=</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/>
<b>hence </b><a NAME="E4:26"/>
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</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>,the <a href="rlvect_1.html#U1">add</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>,the <a href="struct_0.html#U2">Zero</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>,the <a href="vectsp_1.html#U2">lmult</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> #) <a href="hidden.html#R1">=</a> <font color="Maroon">W</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">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_5.html#D4">VECTSP_5:def 4</a><br/>

<b>assume </b><a NAME="E5:26"/>
<font color="Maroon">W</font> <a href="vectsp_5.html#K2">/\</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">y</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#K1">(0).</a> <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>then consider </b><font color="Maroon">z</font> being   <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> such that </b><br/><a NAME="E7:26"/><i><font color="Green">E42</font></i>: 
( ( <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2">/\</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">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; not <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <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> ) or ( <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <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> &amp; not <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2">/\</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">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( ( <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2">/\</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">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; not <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <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> ) or ( not <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2">/\</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">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <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> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:26_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2">/\</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">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E2:26_1_1"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <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/><div class="add"><a NAME="E3:26_1_1"/><i><font color="Green">E68</font></i>: 
<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="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/><a NAME="E4:26_1_1"/>
<font color="Maroon">z</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>, <a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E6:26_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <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_1.html#T60">VECTSP_1:60</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font>
<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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <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="E3:26_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:26_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:26_1_2"/><i><font color="Green">E75</font></i>: 
not <font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K2">/\</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">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E2:26_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <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/><div class="add"><a NAME="E3:26_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <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#T46">VECTSP_4:46</a></i>;<br/><b>hence </b><a NAME="E4:26_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
