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

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:9_1_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_1"><i><font color="Green">E6</font></i></a>, <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">E8</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:9_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:9_1_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E5:9_1_1_1"><i><font color="Green">E8</font></i></a>, <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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></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="vectsp_8.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:9_1_1"/><b>then </b><i><font color="Green">E6</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E6:9_1_1"/>
<font color="Maroon">c<sub>6</sub></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">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:9_1_1_2_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:9_1_1_2_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></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">c<sub>9</sub></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">E10</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:9_1_1_2_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_1_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:9_1_1_2_1_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_1_1"><i><font color="Green">E10</font></i></a>, <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">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E5:9_1_1_2_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E8:9_1_1_2_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="vectsp_8.html#E6:9_1_1_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E10:9_1_1_2_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E8:9_1_1_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:9_1_1_2_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E10:9_1_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="vectsp_8.html#E12:9_1_1_2_1_1"><i><font color="Green">E16</font></i></a>, <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">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E6:9_1_1_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="vectsp_8.html#E12:9_1_1_2_1_1"><i><font color="Green">E16</font></i></a>, <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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></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="vectsp_8.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_1"><i><font color="Green">E9</font></i></a>, <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">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:9_1_1_2_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:9_1_1_2_2"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></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">c<sub>9</sub></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">E10</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:9_1_1_2_2_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_2_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:9_1_1_2_2_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2_2_1"><i><font color="Green">E10</font></i></a>, <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">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E5:9_1_1_2_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E8:9_1_1_2_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="vectsp_8.html#E6:9_1_1_2_2_1"><i><font color="Green">E13</font></i></a>, <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">c<sub>7</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></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">c<sub>7</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E6:9_1_1_2_2_1"><i><font color="Green">E13</font></i></a>, <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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></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="vectsp_8.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E2:9_1_1_2_2"><i><font color="Green">E9</font></i></a>, <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">c<sub>6</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:9_1_1_2"><i><font color="Green">E7</font></i></a>, <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">b<sub>1</sub></font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E4:9_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>


</div>
