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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green" title="E2">A1</font></i>: 
( <font color="Maroon" title="c1">R</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/><b>assume </b><a NAME="E1:5_1_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c2">a</font>
 ;<br/><b>thus </b><font color="Maroon" title="c3">b</font> = 
<font color="Maroon" title="c3">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E1:5_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E1:5_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E1:5_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="binom.html#E1:5_1_1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E1:5_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E1:5_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="binom.html#E1:5_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_1_1"/>
<font color="Maroon" title="c1">R</font> is <a href="algstr_1.html#V3" title="ALGSTR_1:attr.3">add-right-cancelable</a>
 <b>by </b><i><a class="ref" href="algstr_1.html#D7" title="ALGSTR_1:def.7">ALGSTR_1:def 7</a></i>;<br/>


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