<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c6">a1</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:42_2_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="lmod_7.html#T19" target="_self" title="LMOD_7:th.19">Th19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">a2</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:42_2_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="lmod_7.html#T19" target="_self" title="LMOD_7:th.19">Th19</a></i>;<br/>
<div><i><font color="Green" title="E18">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:42_2_1_1"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <b>and </b><br/><a NAME="E2:42_2_1_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 ;<br/><a NAME="E3:42_2_1_1"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E2:42_2_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="lmod_7.html#E1:42_2_1_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="lmod_7.html#T20" target="_self" title="LMOD_7:th.20">Th20</a></i>;<br/><a NAME="E4:42_2_1_1"/>
<font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">b2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E4:42_2_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="lmod_7.html#E2:42_2_1_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="lmod_7.html#T20" target="_self" title="LMOD_7:th.20">Th20</a></i>;<br/><a NAME="E5:42_2_1_1"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">b2</font></span>)</span> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E3:42_2_1_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">b2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">b2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">b2</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T64" title="VECTSP_1:th.64">VECTSP_1:64</a></i>
;<br/><b>hence </b><a NAME="E7:42_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">b2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E5:42_2_1_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="lmod_7.html#T20" target="_self" title="LMOD_7:th.20">Th20</a></i>;<br/></div><b>end;</b></div>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
;<br/>

<b>thus </b><a NAME="E6:42_2_1"/>
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E5:42_2_1"><i><font color="Green" title="E18">A8</font></i></a></i>;<br/>


</div>
