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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">L</font> is <a href="algstr_1.html#V6" target="_self">Loop-like</a>
 ;<br/><a NAME="E2:14_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">L</font> is <a href="algstr_1.html#V2" target="_self">add-left-cancelable</a> &amp; <font color="Maroon">L</font> is <a href="algstr_1.html#V3" target="_self">add-right-cancelable</a> &amp; <font color="Maroon">L</font> is <a href="algstr_1.html#V4" target="_self">add-left-invertible</a> &amp; <font color="Maroon">L</font> is <a href="algstr_1.html#V5" target="_self">add-right-invertible</a> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:14_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:14_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E5:14_1"/>
for <font color="Olive">a</font>, <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>thus </b><a NAME="E6:14_1"/>
( <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14"/>
( ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">a</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) )
 ;<br/>

<b>hence </b><a NAME="E3:14"/>
( <font color="Maroon">L</font> is <a href="algstr_1.html#V2" target="_self">add-left-cancelable</a> &amp; <font color="Maroon">L</font> is <a href="algstr_1.html#V3" target="_self">add-right-cancelable</a> &amp; <font color="Maroon">L</font> is <a href="algstr_1.html#V4" target="_self">add-left-invertible</a> &amp; <font color="Maroon">L</font> is <a href="algstr_1.html#V5" target="_self">add-right-invertible</a> )
 <b>by </b><i>, , , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_1.html#D10" target="_self">ALGSTR_1:def 10</a><br/>


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