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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V2">unital</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">H</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="monoid_0.html#M2">SubStr</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E43</font></i>: 
 <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H</font>)
 ;<br/>

<b>then reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font>, <font color="Maroon">H'</font> = <font color="Maroon">H</font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V2">unital</a>  <a href="group_1.html#L1">HGrStr</a>  <b>by </b><i><a class="ref" href="monoid_0.html#T32">MONOID_0:32</a></i>;<br/>
<a NAME="E3:43"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K11" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">X</font>) <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">G</font>) <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">X</font> &amp; the <a href="vectsp_1.html#U1">unity</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K11" target="_self">.:</a> <font color="Maroon">G'</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="monoid_1.html#K5" target="_self">--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>)</span>)</span> &amp; the <a href="vectsp_1.html#U1">unity</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K11" target="_self">.:</a> <font color="Maroon">H'</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="monoid_1.html#K5" target="_self">--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>)</span>)</span> &amp;  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H'</font>) <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G'</font>) &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">H</font>,<font color="Maroon">X</font>) <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H</font>) <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , , <a class="ref" href="monoid_0.html#T32">MONOID_0:32</a>, <a class="ref" href="monoid_0.html#D23">MONOID_0:def 23</a></i>;<br/>
<a NAME="E4:43"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">H</font>,<font color="Maroon">X</font>) <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K11" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">X</font>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:43"/>
 <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">H</font>,<font color="Maroon">X</font> is    <a href="monoid_0.html#M3">MonoidalSubStr</a> of  <a href="monoid_1.html#K11" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="monoid_0.html#D25">MONOID_0:def 25</a></i>;<br/>


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