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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="algstr_0.html#V5" title="ALGSTR_0:attr.5">left_add-cancelable</a> implies <font color="Maroon" title="c1">R</font> is <a href="algstr_0.html#V6" title="ALGSTR_0:attr.6">right_add-cancelable</a> )</span><br/>

<b>assume </b><a NAME="E1:1_2_1"/>
( <font color="Maroon" title="c1">R</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="algstr_0.html#V5" title="ALGSTR_0:attr.5">left_add-cancelable</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">R</font> is <a href="algstr_0.html#V6" title="ALGSTR_0:attr.6">right_add-cancelable</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">R</font> = <font color="Maroon" title="c1">R</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V5" title="ALGSTR_0:attr.5">left_add-cancelable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a>  ;<br/>
<a NAME="E3:1_2_1"/>
<font color="Maroon" title="c2">R</font> is <a href="algstr_0.html#V6" title="ALGSTR_0:attr.6">right_add-cancelable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D4" title="ALGSTR_0:def.4">ALGSTR_0:def 4</a>,<a class="ref" href="algstr_0.html#D7" title="ALGSTR_0:def.7">ALGSTR_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">a</font> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> )</span><br/>

<b>assume </b><a NAME="E1:1_2_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font></span><br/>

<b>hence </b><a NAME="E2:1_2_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="ref" href="algstr_0.html#D3" title="ALGSTR_0:def.3">ALGSTR_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_2_1"/>
<font color="Maroon" title="c1">R</font> is <a href="algstr_0.html#V6" title="ALGSTR_0:attr.6">right_add-cancelable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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