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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V2" target="_self">add-left-cancelable</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V3" target="_self">add-right-cancelable</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V4" target="_self">add-left-invertible</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V5" target="_self">add-right-invertible</a> )
 <b>by </b><i><a class="ref" href="algstr_1.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:14_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="algstr_1.html#D9" target="_self">Def9</a></i>;<br/><b>thus </b><a NAME="E4:14_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E2:14_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="algstr_1.html#D8" target="_self">Def8</a></i>;<br/><b>thus </b><a NAME="E5:14_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E2:14_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="algstr_1.html#D6" target="_self">Def6</a></i>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E6:14_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="algstr_1.html#E2:14_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="algstr_1.html#D7" target="_self">Def7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) )
 ;<br/>

<b>hence </b><a NAME="E3:14"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V2" target="_self">add-left-cancelable</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V3" target="_self">add-right-cancelable</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V4" target="_self">add-left-invertible</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V5" target="_self">add-right-invertible</a> )
 <b>by </b><i><a class="ref" href="algstr_1.html#D6" target="_self">Def6</a>, <a class="ref" href="algstr_1.html#D7" target="_self">Def7</a>, <a class="ref" href="algstr_1.html#D8" target="_self">Def8</a>, <a class="ref" href="algstr_1.html#D9" target="_self">Def9</a></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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