<?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> be    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>thus </b><a NAME="E1:51"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">c<sub>2</sub></font> #) implies for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</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>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>3</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E2:51_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_2.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:51_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <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>4</sub></font> &amp; <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> )
 <b>by </b><i><a class="txt" href="rmod_3.html#E1:51_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</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>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:51_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of  <a href="rmod_2.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E20"><i><font color="Green">Lemma13</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E2:51_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <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>4</sub></font> &amp; <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> )
 <b>by </b><i><a class="txt" href="rmod_3.html#E2:51"><i><font color="Green">E36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:51_2"/>
<font color="Maroon">c<sub>5</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>4</sub></font>
 <b>by </b><i><a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>by </b><i><a class="ref" href="rmod_2.html#T40">RMOD_2:40</a></i>;<br/>


</div>
