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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W3</font>, <font color="Maroon" title="c5">W2</font> be    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>;<br/>









<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c3">W1</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c4">W3</font> &amp; <font color="Maroon" title="c5">W2</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c4">W3</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" title="c6">v</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/><b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c5">W2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E34">A2</font></i>: 
( <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W2</font> )
 <b>and </b><br/><a NAME="E4:45_1"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">v2</font>
 <b>by </b><i><a class="ref" href="rmod_3.html#T5" target="_self" title="RMOD_3:th.5">Th5</a></i>;<br/><a NAME="E5:45_1"/>
( <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W3</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W3</font> )
 <b>by </b><i><a class="txt" href="rmod_3.html#E1:45"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="rmod_3.html#E3:45_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="rmod_2.html#T16" title="RMOD_2:th.16">RMOD_2:16</a></i>;<br/><b>hence </b><a NAME="E6:45_1"/>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W3</font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E4:45_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="rmod_2.html#T28" title="RMOD_2:th.28">RMOD_2:28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
<font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c5">W2</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c4">W3</font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T36" title="RMOD_2:th.36">RMOD_2:36</a></i>;<br/>


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