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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="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1"/>
<font color="Maroon" title="c1">R</font> is <a href="binom.html#V1" title="BINOM:attr.1">add-cancelable</a>
 ;<br/>

<a NAME="E2:3_2_1"/><b>then </b>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( ( <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">c</font> implies <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) &amp; ( <font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">a</font> implies <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) )
 
<b>by </b><i><a class="ref" href="binom.html#D3" target="_self" title="BINOM:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:3_2_1"/>
( <font color="Maroon" title="c1">R</font> is <a href="algstr_1.html#V2" title="ALGSTR_1:attr.2">add-left-cancelable</a> &amp; <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#D6" title="ALGSTR_1:def.6">ALGSTR_1:def 6</a>, <a class="ref" href="algstr_1.html#D7" title="ALGSTR_1:def.7">ALGSTR_1:def 7</a></i>;<br/>


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