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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">W1</font> be    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</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="E1:4"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> or <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> or <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 ;<br/><div class="add"><a NAME="E2:4_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</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">V</font></span>)</span> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 <b>by </b><i><a class="ref" href="rmod_2.html#T25">RMOD_2:25</a>, <a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>;<br/><b>hence </b><a NAME="E3:4_1_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 ;<br/><div class="add"><a NAME="E2:4_1_1_2"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i><a class="ref" href="rmod_2.html#T25">RMOD_2:25</a>, <a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>;<br/><b>hence </b><a NAME="E3:4_1_1_2"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 ;<br/>


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