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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">v1</font>, <font color="Maroon" title="c4">v2</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>let </b><font color="Maroon" title="c5">W</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>thus </b><a NAME="E1:81"/>
(  ex <font color="Olive" title="b1">C</font> being   <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Maroon" title="c5">W</font> st <br/>( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) implies <font color="Maroon" title="c3">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c6">C</font> being    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Maroon" title="c5">W</font><b> such that </b><a NAME="E1:81_1"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 ;<br/>

<a NAME="E2:81_1"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> st <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font>
 
<b>by </b><i><a class="ref" href="rmod_2.html#D6" target="_self" title="RMOD_2:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:81_1"/>
<font color="Maroon" title="c3">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W</font>
 <b>by </b><i><a class="txt" href="rmod_2.html#E1:81_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="rmod_2.html#T76" target="_self" title="RMOD_2:th.76">Th76</a></i>;<br/>


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

<b>assume </b><a NAME="E2:81"/>
<font color="Maroon" title="c3">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">v</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="E4:81"/><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> &amp; <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> )
 <b>by </b><i><a class="ref" href="rmod_2.html#T76" target="_self" title="RMOD_2:th.76">Th76</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c6">v</font> <a href="rmod_2.html#K3" title="RMOD_2:func.3">+</a> <font color="Maroon" title="c5">W</font> as    <a href="rmod_2.html#M2" title="RMOD_2:mode.2">Coset</a> of <font color="Maroon" title="c5">W</font> <b>by </b><i><a class="ref" href="rmod_2.html#D6" target="_self" title="RMOD_2:def.6">Def6</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">C</font>
;<br/>

<b>thus </b><a NAME="E6:81"/>
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="txt" href="rmod_2.html#E4:81"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>


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