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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:83"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>
 ;<br/>

<a NAME="E2:83"/><b>then </b><i><font color="Green" title="E46">A2</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; ( 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">G</font>  st ( <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">c</font> or <font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> ) holds <br/><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="monoid_0.html#T15" target="_self" title="MONOID_0:th.15">Th15</a>, <a class="ref" href="monoid_0.html#T25" target="_self" title="MONOID_0:th.25">Th25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>let </b><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">H</font>;<br/><b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">c'</font> = <font color="Maroon" title="c5">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="monoid_0.html#E2:83"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:83_1"/>
( <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b'</font> &amp; <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">c'</font> &amp; <font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a'</font> &amp; <font color="Maroon" title="c5">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a'</font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E3:83_1"/>
( ( <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font> or <font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font> ) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:83"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="monoid_0.html#T15" target="_self" title="MONOID_0:th.15">Th15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:83"/>
<font color="Maroon" title="c2">H</font> is <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>
 <b>by </b><i><a class="ref" href="monoid_0.html#T15" target="_self" title="MONOID_0:th.15">Th15</a></i>;<br/>


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