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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_2.html#NM7">RightMod</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="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:45"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:45_1"/>
<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="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green">E34</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> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:45_1"/><i><font color="Green">E35</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>by </b><i><a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E5:45_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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="rmod_3.html#E1:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rmod_3.html#E3:45_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rmod_2.html#T16">RMOD_2:16</a></i>;<br/><b>hence </b><a NAME="E6:45_1"/>
<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="txt" href="rmod_3.html#E4:45_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rmod_2.html#T28">RMOD_2:28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T36">RMOD_2:36</a></i>;<br/>


</div>
