<?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> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:47"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlsub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:47"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlsub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a>, <a class="ref" href="rlsub_2.html#T22" target="_self">Th22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green">E39</font></i></a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E1:47_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlsub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E2:47_1"/>
<font color="Maroon">c<sub>5</sub></font> is  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Olive">b<sub>2</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E7:47"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2">RELSET_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E8:47"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E2:47_2_1"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T33">RLSUB_1:33</a></i>;<br/><b>hence </b><a NAME="E3:47_2_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:47"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:47_3_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:47_3_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E3:47_3_1"/><i><font color="Green">E46</font></i>: 
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_3_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E4:47_3_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rlsub_2.html#E2:47_3_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E5:47_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_3_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rlsub_1.html#T34">RLSUB_1:34</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:47"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:47_4_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:47_4_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> <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>, <font color="Maroon">c<sub>10</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:47_4_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_4_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:47_4_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rlsub_2.html#E2:47_4_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E7:47_4_1"/>
<font color="Maroon">c<sub>9</sub></font> is  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_4_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_4_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rlsub_1.html#T35">RLSUB_1:35</a></i>;<br/><b>hence </b><a NAME="E8:47_4_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rlsub_2.html#E4:47_4_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_4_1"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_4"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:47"/><i><font color="Green">E45</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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp;  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5"><b>proof </b></a><div class="add">

<a NAME="E1:47_5"/>
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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_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><b>that </b><br/><a NAME="E1:47_5_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:47_5_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</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> &amp; not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_5_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E3:47_5_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>hence </b><a NAME="E4:47_5_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> );<br/><a NAME="E2:47_5_1_1_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/><a NAME="E3:47_5_1_1_1_2"/><i><font color="Green">E50</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> <font color="Maroon">c<sub>6</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_5_1_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:47_5_1_1_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E4:47_5_1_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_5_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E6:47_5_1_1_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</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="rlsub_2.html#E3:47_5_1_1_1_2_1"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:47_5_1_1_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:47_5_1_1_1_2"/><i><font color="Green">E52</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> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="rlsub_2.html#E2:47_5_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rlsub_2.html#E3:47_5_1_1_1_2"><i><font color="Green">E50</font></i></a>);<br/></i><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_2"><b>now </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="E1:47_5_1_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_5_1_1_1_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:47_5_1_1_1_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:47_5_1_1_1_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E7:47_5_1_1_1_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:47_5_1_1_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E10:47_5_1_1_1_2_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_2"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E11:47_5_1_1_1_2_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_5_1_1_1_2_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><b>hence </b><a NAME="E12:47_5_1_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:47_5_1_1_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</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="rlsub_2.html#E1:47_5_1_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:47_5_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E13:47_5_1_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E9:47_5_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:47_5_1_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlsub_2.html#E12:47_5_1_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E16:47_5_1_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</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="rlsub_2.html#E9:47_5_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rlsub_2.html#E15:47_5_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="orders_1.html#T91">ORDERS_1:91</a></i>;<br/><a NAME="E17:47_5_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_5_1_1_1_2_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</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>9</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_5_1_1_1_2_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E2:47_5_1_1_1_2_3_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:47_5_1_1_1_2_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E5:47_5_1_1_1_2_3_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1_1_1_2_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:47_5_1_1_1_2_3_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E8:47_5_1_1_1_2_3_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47_5_1_1_1_2_3_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:47_5_1_1_1_2_3_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E11:47_5_1_1_1_2_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:47_5_1_1_1_2_3_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E14:47_5_1_1_1_2_3_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_3_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:47_5_1_1_1_2_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E17:47_5_1_1_1_2_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:47_5_1_1_1_2_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E20:47_5_1_1_1_2_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_5_1_1_1_2_3_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="rlsub_2.html#E2:47_5_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green">E66</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_2_3_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E3:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47_5_1_1_1_2_3_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="rlsub_2.html#E19:47_5_1_1_1_2_3_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_3_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_3_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rlsub_2.html#E14:47_5_1_1_1_2_3_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="rlsub_2.html#E17:47_5_1_1_1_2_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="rlsub_2.html#E20:47_5_1_1_1_2_3_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T28">RLSUB_1:28</a></i>;<br/><a NAME="E6:47_5_1_1_1_2_3_1_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:47_5_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_3_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:47_5_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</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="rlsub_2.html#E20:47_5_1_1_1_2_3_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_3_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_5_1_1_1_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_5_1_1_1_2_3_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E3:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47_5_1_1_1_2_3_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="rlsub_2.html#E19:47_5_1_1_1_2_3_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_3_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_3_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rlsub_2.html#E14:47_5_1_1_1_2_3_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="rlsub_2.html#E17:47_5_1_1_1_2_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="rlsub_2.html#E20:47_5_1_1_1_2_3_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T28">RLSUB_1:28</a></i>;<br/><a NAME="E6:47_5_1_1_1_2_3_1_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:47_5_1_1_1_2_3_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E10:47_5_1_1_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:47_5_1_1_1_2_3_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</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="rlsub_2.html#E17:47_5_1_1_1_2_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_3_1_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E23:47_5_1_1_1_2_3_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E2:47_5_1_1_1_2_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:47_5_1_1_1_2_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E5:47_5_1_1_1_2_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:47_5_1_1_1_2_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E8:47_5_1_1_1_2_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:47_5_1_1_1_2_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E11:47_5_1_1_1_2_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E12:47_5_1_1_1_2_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_5_1_1_1_2_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:47_5_1_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T29">RLSUB_1:29</a></i>;<br/>
<a NAME="E14:47_5_1_1_1_2_3"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E15:47_5_1_1_1_2_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:47_5_1_1_1_2_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:47_5_1_1_1_2_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</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="rlsub_2.html#E11:47_5_1_1_1_2_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rlsub_2.html#E14:47_5_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:47_5_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47_5_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rlsub_1.html#T43">RLSUB_1:43</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:47_5_1_1_1_2_4"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>13</sub></font> implies <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_5_1_1_1_2_4_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:47_5_1_1_1_2_4_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:47_5_1_1_1_2_4_1"/><b>then </b>
<font color="Maroon">c<sub>14</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="rlsub_2.html#E19:47_5_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:47_5_1_1_1_2_4_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E6:47_5_1_1_1_2_4_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:47_5_1_1_1_2_4_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E9:47_5_1_1_1_2_4_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2_4_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:47_5_1_1_1_2_4_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E12:47_5_1_1_1_2_4_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_4_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E13:47_5_1_1_1_2_4_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_5_1_1_1_2_4_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2_4_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rlsub_2.html#E9:47_5_1_1_1_2_4_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="rlsub_2.html#E12:47_5_1_1_1_2_4_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E14:47_5_1_1_1_2_4_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rlsub_2.html#E1:47_5_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_5_1_1_1_2_4_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:47_5_1_1_1_2_4_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E11:47_5_1_1_1_2_4_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rlsub_2.html#E13:47_5_1_1_1_2_4_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:47_5_1_1_1_2_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:47_5_1_1_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_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#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D3">RLSUB_1:def 3</a></i>;<br/><a NAME="E5:47_5_1_1_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:47_5_1_1_1_2_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E21:47_5_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T39">RLSUB_1:39</a></i>;<br/><a NAME="E22:47_5_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlsub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rlsub_2.html#E21:47_5_1_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E24:47_5_1_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E26:47_5_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E27:47_5_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_5_1_1_1_2"><i><font color="Green">E52</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5_1_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:47_5_1_1_1_2_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E2:47_5_1_1_1_2_5"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:47_5_1_1_1_2_5"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_5_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rlsub_2.html#E24:47_5_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rlsub_2.html#E27:47_5_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:47_5_1_1_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>18</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="rlsub_2.html#E2:47_5_1_1_1_2_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E5:47_5_1_1_1_2_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E19:47_5_1_1_1_2"><i><font color="Green">E58</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="E29:47_5_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> is  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T37">RLSUB_1:37</a></i>;<br/><b>hence </b><a NAME="E30:47_5_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rlsub_2.html#E26:47_5_1_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_5_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</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/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <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="E2:47_5"/>
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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp;  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E13:47"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1">ORDERS_1:sch 1</a>(<a class="txt" href="rlsub_2.html#E8:47"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rlsub_2.html#E9:47"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rlsub_2.html#E10:47"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rlsub_2.html#E11:47"><i><font color="Green">E45</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:47"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E16:47"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E17:47"/>
 <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_2.html#D4" target="_self">RLSUB_2:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_6"/>
not  <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 ;<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="E3:47_6"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_2.html#E56"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E4:47_6"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rlsub_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : verum </span>}</span> ;<br/>
<a NAME="E5:47_6"/><i><font color="Green">E51</font></i>: 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_6_1"/>
<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="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : verum </span>}</span> 
 ;<br/><a NAME="E2:47_6_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>hence </b><a NAME="E3:47_6_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:47_6"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_6_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</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>10</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:47_6_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:47_6_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>
<b>assume </b><a NAME="E4:47_6_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:47_6_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>
<a NAME="E7:47_6_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_6_2_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rlsub_2.html#E6:47_6_2_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:47_6_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E2:47_6_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:47_6_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>
<a NAME="E5:47_6_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_6_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:47_6_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:47_6"/><i><font color="Green">E52</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlsub_1.html#T43">RLSUB_1:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:47_6_3"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> implies <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_6_3_1"/><i><font color="Green">E53</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> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:47_6_3_1"/><b>then </b>
<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="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:47_6_3_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E10:47_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:47_6_3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_6_3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:47_6_3_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_6_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E3:47_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rlsub_1.html#T29">RLSUB_1:29</a></i>;<br/><a NAME="E4:47_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E5:47_6_3_1_1"/><b>then </b>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_6_3_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E6:47_6_3_1_1"/><b>then </b>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E10"><i><font color="Green">Lemma6</font></i></a></i>;<br/><b>hence </b><a NAME="E7:47_6_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:47_6_3_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E8:47_6_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:47_6_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:47_6_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_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#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:47_6_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D3">RLSUB_1:def 3</a></i>;<br/><a NAME="E5:47_6_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:47_6_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:47_6"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T39">RLSUB_1:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:47_6_4"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_6_4_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:47_6_4_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</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="E4:47_6_4_1"/><i><font color="Green">E55</font></i>: 
<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>and </b><br/><a NAME="E5:47_6_4_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E6:47_6_4_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E7:47_6_4_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_6_4_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rlsub_2.html#E57"><i><font color="Green">Lemma37</font></i></a></i>;<br/>
<a NAME="E8:47_6_4_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:47_6_4_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E9:47_6_4_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6_4_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rlsub_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:47_6_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:47_6_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rlsub_1.html#T31">RLSUB_1:31</a></i>;<br/>
<a NAME="E11:47_6_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E4:47_6_4_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E12:47_6_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E12:47_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:47_6_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D3">RLSUB_1:def 3</a></i>;<br/>
<a NAME="E14:47_6_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:47_6_4_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E6:47_6_4_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E16:47_6_4_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rlsub_2.html#E5:47_6_4_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rlsub_2.html#E8:47_6_4_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:47_6_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:47_6_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_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#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:47_6_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D3">RLSUB_1:def 3</a></i>;<br/><a NAME="E5:47_6_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:47_6_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:47_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T39">RLSUB_1:39</a></i>;<br/>
<a NAME="E15:47_6"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlsub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a>, <a class="ref" href="rlsub_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="rlsub_2.html#E2:47"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E17:47_6"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</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> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</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> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E18:47_6"/><b>then </b><i><font color="Green">E54</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E3:47_6"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E19:47_6"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E5:47_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E20:47_6"/><b>then </b>
<font color="Maroon">c<sub>8</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="rlsub_2.html#E10:47_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E21:47_6"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E18:47_6"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rlsub_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E22:47_6"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E23:47_6"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E7:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rlsub_2.html#E15:47"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E24:47_6"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_2.html#E13:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rlsub_2.html#E15:47"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rlsub_2.html#E21:47_6"><i><font color="Green">E55</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E18:47"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E16:47"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rlsub_2.html#T18" target="_self">Th18</a></i>;<br/>


</div>
