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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">V1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">V2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">V3</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">V1</font> is <a href="rmod_2.html#V1" target="_self">lineary-closed</a> &amp; <font color="Maroon">V2</font> is <a href="rmod_2.html#V1" target="_self">lineary-closed</a> )
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">V3</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E3:7"/>
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rmod_2.html#D1" target="_self">RMOD_2:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:7_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font>
 <b>and </b><br/><a NAME="E2:7_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font>
 ;<br/>

<b>consider </b><font color="Maroon">v1</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:7_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>and </b><br/><a NAME="E5:7_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">u1</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">u2</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:7_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font>
 <b>and </b><br/><a NAME="E8:7_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:7_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 
<b>by </b><i>, , , </i>;<br/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</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">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</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">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E11:7_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font>
 <b>by </b><i>, </i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E4:7"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v1</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:7"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>and </b><br/><a NAME="E7:7"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:7"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">v1</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">v2</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:7"/>
<font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_2.html#D23">VECTSP_2:def 23</a></i>;<br/>
<b>hence </b><a NAME="E10:7"/>
<font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V3</font>
 <b>by </b><i>, </i>;<br/>


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