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<span class="kw">let </span><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" 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="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/>

<span class="kw">assume </span><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" 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="c1">R</font> is  <a href="algstr_0.html#V6" title="ALGSTR_0:attr.6">right_add-cancelable</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D4" onmouseover="rs('algstr_0/D4')" onmouseout="rh()">ALGSTR_0:def 4</a>,<a class="ref" href="algstr_0.html#D7" onmouseover="rs('algstr_0/D7')" onmouseout="rh()">ALGSTR_0:def 7</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"> (  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/>

<span class="kw">assume </span><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" 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="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font></span><br/>

<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_0.html#D3" onmouseover="rs('algstr_0/D3')" onmouseout="rh()">ALGSTR_0:def 3</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="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" 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/>


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