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

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E1:5_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:5_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:5_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectsp_6.html#K2">ZeroLC</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#T26">VECTSP_6:26</a></i>;<br/>
<a NAME="E4:5_1_1"/>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T41">VECTSP_6:41</a></i>;<br/>
<a NAME="E5:5_1_1"/><b>then </b><i><font color="Green">E5</font></i>: 
 <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>5</sub></font>
 
;<br/>
<a NAME="E6:5_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:5_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="vectsp_1.html#NM5">Vector</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>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</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>5</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_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="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:5_1_1_2_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:5_1_1_2_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>5</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:5_1_1_2_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="mod_3.html#E1:5_1_1_2_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:5_1_1_2_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mod_3.html#E2:5_1_1_2_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">c<sub>10</sub></font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#T52">VECTSP_6:52</a></i>;<br/>
<a NAME="E8:5_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#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="mod_3.html#E4:5_1_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="mod_3.html#E6:5_1_1_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="vectsp_6.html#T77">VECTSP_6:77</a></i>;<br/>
<b>hence </b><a NAME="E9:5_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>5</sub></font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="vectsp_1.html#NM3">Scalar</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="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:5_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:5_1_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#T61">VECTSP_6:61</a></i>;<br/>
<a NAME="E6:5_1_1_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#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="mod_3.html#E4:5_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="mod_3.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:5_1_1_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>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:5_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 the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="mod_3.html#E5:5_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>


</div>
