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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">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>thus </b><a NAME="E1:51"/>
( <font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_2.html#G1" title="VECTSP_2:aggr.1">RightModStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c2">V</font> #) implies for <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>  ex <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b2">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="Olive" title="b3">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b3">v2</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" title="E36">A1</font></i>: 
<font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_2.html#G1" title="VECTSP_2:aggr.1">RightModStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c2">V</font> #)
 ;<br/>

<b>let </b><font color="Maroon" title="c5">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/>

<a NAME="E2:51_1"/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="rmod_2.html#K2" title="RMOD_2:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:51_1"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">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="Olive" title="b2">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">v2</font> )
 <b>by </b><i><a class="txt" href="rmod_3.html#E1:51_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="rmod_3.html#T5" target="_self" title="RMOD_3:th.5">Th5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:51"/><i><font color="Green" title="E36">A2</font></i>: 
for <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>  ex <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b2">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="Olive" title="b3">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b3">v2</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" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of  <a href="rmod_2.html#K2" title="RMOD_2:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E20"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c5">u</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/><a NAME="E2:51_2"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">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="Olive" title="b2">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">v2</font> )
 <b>by </b><i><a class="txt" href="rmod_3.html#E2:51"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:51_2"/>
<font color="Maroon" title="c5">u</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="c4">W2</font>
 <b>by </b><i><a class="ref" href="rmod_3.html#T5" target="_self" title="RMOD_3:th.5">Th5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
<font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_2.html#G1" title="VECTSP_2:aggr.1">RightModStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c2">V</font> #)
 <b>by </b><i><a class="ref" href="rmod_2.html#T40" title="RMOD_2:th.40">RMOD_2:40</a></i>;<br/>


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