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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:65"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:65"/><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>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:65"/><i><font color="Green">E50</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/>
<b>assume </b><a NAME="E4:65"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:65"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E1:65"><i><font color="Green">E48</font></i></a>, <a class="ref" href="vectsp_5.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T37">VECTSP_4:37</a>, <a class="ref" href="vectsp_4.html#T50">VECTSP_4:50</a></i>;<br/>
<a NAME="E7:65"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <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>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E3:65"><i><font color="Green">E50</font></i></a>, <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<a NAME="E8:65"/><b>then </b>
<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>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R2">c&lt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:65"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>6</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"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:65"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">c<sub>6</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>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T77">RLSUB_2:77</a></i>;<br/>
<a NAME="E12:65"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E10:65"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:65"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E14:65"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E11:65"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:65"/>
<font color="Maroon">c<sub>6</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="vectsp_5.html#E12:65"><i><font color="Green">E53</font></i></a>, <a class="ref" href="vectsp_4.html#T17">VECTSP_4:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E18:65"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E19:65"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E1:65"><i><font color="Green">E48</font></i></a>, <a class="txt" href="vectsp_5.html#E64"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <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>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_5.html#E19:65"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E21:65"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E13:65"><i><font color="Green">E54</font></i></a>, <a class="txt" href="vectsp_5.html#E18:65"><i><font color="Green">E56</font></i></a>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a>, <a class="ref" href="vectsp_4.html#T31">VECTSP_4:31</a></i>;<br/>
<a NAME="E22:65"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:65"><i><font color="Green">E49</font></i></a>, <a class="txt" href="vectsp_5.html#E18:65"><i><font color="Green">E56</font></i></a>, <a class="txt" href="vectsp_5.html#E19:65"><i><font color="Green">E57</font></i></a>, <a class="txt" href="vectsp_5.html#E20:65"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>9</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>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E24:65"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>
<a NAME="E25:65"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E26:65"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_5.html#E14:65"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>;<br/>


</div>
