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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">L</font> being    <a href="rmod_4.html#M1" title="RMOD_4:mode.1">Linear_Combination</a> of <font color="Olive" title="b1">V</font> holds <br/> ( <font color="Olive" title="b2">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Olive" title="b2">L</font> &amp; <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Olive" title="b2">L</font> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Olive" title="b2">L</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being    <a href="rmod_4.html#M1" title="RMOD_4:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> holds <br/> ( <font color="Olive" title="b1">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Olive" title="b1">L</font> &amp; <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Olive" title="b1">L</font> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Olive" title="b1">L</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be    <a href="rmod_4.html#M1" title="RMOD_4:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Maroon" title="c3">L</font> &amp; <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c3">L</font> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Maroon" title="c3">L</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:49"/>
<font color="Maroon" title="c3">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Maroon" title="c3">L</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c3">L</font> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Maroon" title="c3">L</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c2">V</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rmod_4.html#D8" target="_self" onmouseover="rs('rmod_4/D8')" onmouseout="rh()">RMOD_4:def 8</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#D9" target="_self" onmouseover="rs('rmod_4/D9')" onmouseout="rh()">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#T18" target="_self" onmouseover="rs('rmod_4/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">hence </span><a NAME="E2:49"/>
<span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c3">L</font> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#T39" target="_self" onmouseover="rs('rmod_4/T39')" onmouseout="rh()">Th39</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
