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

<b>reconsider </b><font color="Maroon" title="c4">cG</font> = <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">VS</font> ;<br/>
<b>set </b><font color="Maroon" title="c5">C0</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">cG</font>;<br/>
<a NAME="E2:9_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font> holds <br/> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">VS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">VS</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">X1</font> being    <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">X1</font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:9_1_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">X1</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:9_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self" title="VECTSP_8:def.4">Def4</a></i>;<br/>
<a NAME="E6:9_1_1_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">VS</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c8">X1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">VS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E5:9_1_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">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" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="vectsp_8.html#T1" target="_self" title="VECTSP_8:th.1">Th1</a></i>;<br/>
<a NAME="E4:9_1_1"/><b>then </b><i><font color="Green" title="E6">A5</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">cG</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">C0</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">cG</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">VS</font> ;<br/>
<a NAME="E6:9_1_1"/>
<font color="Maroon" title="c6">C0</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">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" title="E7">A6</font></i>: 
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">VS</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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" title="c7">v</font>, <font color="Maroon" title="c8">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">VS</font>;<br/>

<b>assume </b><a NAME="E1:9_1_1_2_1"/><i><font color="Green" title="E8">A7</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font> )
 ;<br/>

<a NAME="E2:9_1_1_2_1"/><i><font color="Green" title="E9">A8</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font> holds <br/><font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c9">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_1_2_1_1"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<a NAME="E2:9_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E11">A10</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> )
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c9">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">VS</font> <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_1_1"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">X1</font> being    <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E5:9_1_1_2_1_1"/><i><font color="Green" title="E12">A11</font></i>: 
( <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">X1</font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:9_1_1_2_1_1"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">X1</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E5:9_1_1_2_1_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self" title="VECTSP_8:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c13">u1</font> = <font color="Maroon" title="c8">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">VS</font> ;<br/>
<a NAME="E8:9_1_1_2_1_1"/><i><font color="Green" title="E14">A13</font></i>: 
( <font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">X1</font> &amp; <font color="Maroon" title="c13">u1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">X1</font> )
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="vectsp_8.html#E6:9_1_1_2_1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">vu</font> = <font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">u1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">VS</font> ;<br/>
<a NAME="E10:9_1_1_2_1_1"/><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c14">vu</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">X1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c11">X1</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E8:9_1_1_2_1_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">X2</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E12:9_1_1_2_1_1"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c15">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E13:9_1_1_2_1_1"/>
<font color="Maroon" title="c14">vu</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c15">X2</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E10:9_1_1_2_1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="vectsp_8.html#E12:9_1_1_2_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="vectsp_5.html#T8" title="VECTSP_5:th.8">VECTSP_5:8</a></i>;<br/>
<b>hence </b><a NAME="E14:9_1_1_2_1_1"/>
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E6:9_1_1_2_1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="vectsp_8.html#E12:9_1_1_2_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">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" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="vectsp_8.html#T1" target="_self" title="VECTSP_8:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_2_1"/>
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9_1_1_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">VS</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font> holds <br/><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c8">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">VS</font>;<br/>



<b>assume </b><a NAME="E1:9_1_1_2_2"/><i><font color="Green" title="E8">A16</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font>
 ;<br/>

<a NAME="E2:9_1_1_2_2"/><i><font color="Green" title="E9">A17</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font> holds <br/><font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c9">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_1_2_2_1"/><i><font color="Green" title="E10">A18</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<a NAME="E2:9_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E11">A19</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_2"><i><font color="Green" title="E8">A16</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c9">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">VS</font> <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_2_1"><i><font color="Green" title="E10">A18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">X1</font> being    <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E5:9_1_1_2_2_1"/><i><font color="Green" title="E12">A20</font></i>: 
( <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">X1</font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_2_1"><i><font color="Green" title="E10">A18</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:9_1_1_2_2_1"/><i><font color="Green" title="E13">A21</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">X1</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E5:9_1_1_2_2_1"><i><font color="Green" title="E12">A20</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self" title="VECTSP_8:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c8">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">VS</font> ;<br/>
<a NAME="E8:9_1_1_2_2_1"/>
<font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">X1</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_2_1"><i><font color="Green" title="E11">A19</font></i></a>, <a class="txt" href="vectsp_8.html#E6:9_1_1_2_2_1"><i><font color="Green" title="E13">A21</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E9:9_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">X1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></i>;<br/>
<b>hence </b><a NAME="E10:9_1_1_2_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E6:9_1_1_2_2_1"><i><font color="Green" title="E13">A21</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">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" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="vectsp_8.html#T1" target="_self" title="VECTSP_8:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C0</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_2"><i><font color="Green" title="E9">A17</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">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" title="c6">C0</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">lineary-closed</a>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">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" title="b1">W</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font> st  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">W</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E4:9_1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="vectsp_4.html#T42" title="VECTSP_4:th.42">VECTSP_4:42</a></i>;<br/>


</div>
