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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> ;<br/>
<a NAME="E3:60"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E1:60"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:60"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E6:60"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E7:60"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E2:60"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:60"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E10:60"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<b>thus </b><a NAME="E11:60"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:60_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:60_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:60_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:60_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E5:60"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>then </b><i><font color="Green">E53</font></i>: <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E9:60"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rlsub_1.html#E4:60_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E8:60_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:60"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rlsub_1.html#E10:60"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rlsub_1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E9:60_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E5:60_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rlsub_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E10:60_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E7:60_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:60_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>10</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="E12:60"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E13:60"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:60"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E16:60"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E9:60"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>then </b><i><font color="Green">E53</font></i>: <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="rlsub_1.html#E5:60"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rlsub_1.html#E15:60"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E19:60"/>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E6:60"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rlsub_1.html#E10:60"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rlsub_1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E20:60"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E16:60"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rlsub_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E21:60"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E18:60"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:60"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div>
