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

<b>reconsider </b><font color="Maroon">cG</font> = <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">VS</font> ;<br/>
<b>set </b><font color="Maroon">C0</font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">cG</font>;<br/>
<a NAME="E2:9_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> holds <br/> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">VS</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </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="E1:9_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">VS</font> ;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="vectsp_8.html#M2" target="_self">Element</a> of  <a href="vectsp_8.html#K2" target="_self">Subspaces</a> <font color="Maroon">VS</font><b> such that </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:9_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:9_1_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">VS</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">VS</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9_1_1"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:9_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">cG</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C0</font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">cG</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">VS</font> ;<br/>
<a NAME="E6:9_1_1"/>
<font color="Maroon">C0</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:9_1_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font>;<br/>

<b>assume </b><a NAME="E1:9_1_1_2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font> )
 ;<br/>

<a NAME="E2:9_1_1_2_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_1_1"><b>proof </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="E1:9_1_1_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:9_1_1_2_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">VS</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="vectsp_8.html#M2" target="_self">Element</a> of  <a href="vectsp_8.html#K2" target="_self">Subspaces</a> <font color="Maroon">VS</font><b> such that </b><br/><a NAME="E5:9_1_1_2_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:9_1_1_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">u1</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font> ;<br/>
<a NAME="E8:9_1_1_2_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">vu</font> = <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font> ;<br/>
<a NAME="E10:9_1_1_2_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">vu</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/>
<b>consider </b><font color="Maroon">X2</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">VS</font><b> such that </b><br/><a NAME="E12:9_1_1_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E13:9_1_1_2_1_1"/>
<font color="Maroon">vu</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, , <a class="ref" href="vectsp_5.html#T8">VECTSP_5:8</a></i>;<br/>
<b>hence </b><a NAME="E14:9_1_1_2_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:9_1_1_2_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9_1_1_2"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><br/> for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font> holds <br/><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font>;<br/>



<b>assume </b><a NAME="E1:9_1_1_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font>
 ;<br/>

<a NAME="E2:9_1_1_2_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_2_1"><b>proof </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="E1:9_1_1_2_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:9_1_1_2_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">VS</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="vectsp_8.html#M2" target="_self">Element</a> of  <a href="vectsp_8.html#K2" target="_self">Subspaces</a> <font color="Maroon">VS</font><b> such that </b><br/><a NAME="E5:9_1_1_2_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:9_1_1_2_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">VS</font> ;<br/>
<a NAME="E8:9_1_1_2_2_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E9:9_1_1_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<b>hence </b><a NAME="E10:9_1_1_2_2_1"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:9_1_1_2_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C0</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1_1_2"/>
<font color="Maroon">C0</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:9_1_1"/>
 ex <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">VS</font> st  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">VS</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">W</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>


</div>
