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<b>let </b><font color="Maroon" title="c1">L</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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V6" title="ALGSTR_1:attr.6">Loop-like</a>
 ;<br/><a NAME="E2:14_1"/><b>then </b><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V2" title="ALGSTR_1:attr.2">add-left-cancelable</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V3" title="ALGSTR_1:attr.3">add-right-cancelable</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V4" title="ALGSTR_1:attr.4">add-left-invertible</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V5" title="ALGSTR_1:attr.5">add-right-invertible</a> )
 <b>by </b><i><a class="ref" href="algstr_1.html#D10" target="_self" title="ALGSTR_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:14_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 <b>by </b><i><a class="ref" href="algstr_1.html#D9" target="_self" title="ALGSTR_1:def.9">Def9</a></i>;<br/><b>thus </b><a NAME="E4:14_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b3">x</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="b2">b</font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E2:14_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="algstr_1.html#D8" target="_self" title="ALGSTR_1:def.8">Def8</a></i>;<br/><b>thus </b><a NAME="E5:14_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">x</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">y</font> holds <br/><font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E2:14_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="algstr_1.html#D6" target="_self" title="ALGSTR_1:def.6">Def6</a></i>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>thus </b><a NAME="E6:14_1"/>
( <font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="algstr_1.html#E2:14_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="algstr_1.html#D7" target="_self" title="ALGSTR_1:def.7">Def7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14"/>
( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b3">x</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="b2">b</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">x</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">y</font> holds <br/><font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">x</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">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> ) )
 ;<br/>

<b>hence </b><a NAME="E3:14"/>
( <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V2" title="ALGSTR_1:attr.2">add-left-cancelable</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V3" title="ALGSTR_1:attr.3">add-right-cancelable</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V4" title="ALGSTR_1:attr.4">add-left-invertible</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="algstr_1.html#V5" title="ALGSTR_1:attr.5">add-right-invertible</a> )
 <b>by </b><i><a class="ref" href="algstr_1.html#D6" target="_self" title="ALGSTR_1:def.6">Def6</a>, <a class="ref" href="algstr_1.html#D7" target="_self" title="ALGSTR_1:def.7">Def7</a>, <a class="ref" href="algstr_1.html#D8" target="_self" title="ALGSTR_1:def.8">Def8</a>, <a class="ref" href="algstr_1.html#D9" target="_self" title="ALGSTR_1:def.9">Def9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_1.html#D10" target="_self" title="ALGSTR_1:def.10">ALGSTR_1:def 10</a><br/>


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