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

<b>let </b><font color="Maroon">K</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> is <a href="lmod_5.html#V1">linearly-independent</a> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>reconsider </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font></span>)</span> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> ;<br/>
<b>set </b><font color="Maroon">Z</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>;<br/>
<a NAME="E3:16"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D2">VECTSP_5:def 2</a></i>;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 ;<br/><a NAME="E2:16_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:16_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">l1</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A1</font><b> such that </b><br/><a NAME="E5:16_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="mod_3.html#T11">MOD_3:11</a></i>;<br/><b>consider </b><font color="Maroon">l2</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A2</font><b> such that </b><br/><a NAME="E7:16_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l2</font>
 <b>by </b><i>, <a class="ref" href="mod_3.html#T11">MOD_3:11</a></i>;<br/><a NAME="E8:16_1"/><i><font color="Green">E78</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><a NAME="E9:16_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/><a NAME="E10:16_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">l2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T74">VECTSP_6:74</a></i>;<br/><a NAME="E11:16_1"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">l2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l1</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">l2</font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/> <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T80">VECTSP_6:80</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i>, , <a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a></i>
;<br/><a NAME="E14:16_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="lmod_5.html#D1">LMOD_5:def 1</a></i>;<br/><a NAME="E15:16_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l1</font>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:16_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> )
 <b>by </b><i>, <a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/>
<a NAME="E5:16_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:16_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> = 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T73">VECTSP_6:73</a></i>
;<br/>
<b>then </b><font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E9:16_1_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="vectsp_6.html#T45">VECTSP_6:45</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:16"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:16"/>
<span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A1</font></span>)</span> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/>


</div>
