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<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>, <font color="Maroon">c<sub>5</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="E1:36"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:36"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<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="vectsp_5.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1"/>
<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>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:36_1"/><b>then </b><i><font color="Green">E33</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 <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</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#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:36_1"/><b>then </b>
<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#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:36_1"/><b>then </b>
<font color="Maroon">c<sub>6</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#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ( <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>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E6:36_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E7:36_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E8:36_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E9:36_1"/><i><font color="Green">E37</font></i>: 
<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#E1:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp_5.html#E7:36_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="vectsp_4.html#T16">VECTSP_4:16</a></i>;<br/>
<a NAME="E10:36_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</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#E7:36_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E11:36_1"/>
<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 <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:36_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:36_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="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#E6:36_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:36_1"/><b>then </b>
<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#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#E9:36_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="vectsp_4.html#T31">VECTSP_4:31</a></i>;<br/>
<a NAME="E14:36_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <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> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E15:36_1"/><b>then </b>
<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>2</sub></font></span>)</span> <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_1.html#T66">VECTSP_1:66</a></i>;<br/>
<a NAME="E16:36_1"/><b>then </b>
<font color="Maroon">c<sub>8</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="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E17:36_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E8:36_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="vectsp_5.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E18:36_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <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="vectsp_5.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:36_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="vectsp_5.html#E10:36_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="vectsp_5.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E19:36_1"/>
<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"><span class="p2">(<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="vectsp_5.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E3:36"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<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="vectsp_5.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</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>4</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E45"><i><font color="Green">Lemma30</font></i></a></i>;<br/>


</div>
