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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">H</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c2">G</font>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">H</font>)
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">G'</font> = <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c5">H'</font> = <font color="Maroon" title="c3">H</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  <b>by </b><i><a class="ref" href="monoid_0.html#T32" title="MONOID_0:th.32">MONOID_0:32</a></i>;<br/>
<a NAME="E3:43"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H</font>) <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>) &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K11" title="MONOID_1:func.11">.:</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">X</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">G</font>) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K11" title="MONOID_1:func.11">.:</a> <font color="Maroon" title="c4">G'</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>)</span>)</span> &amp; the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K11" title="MONOID_1:func.11">.:</a> <font color="Maroon" title="c5">H'</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H</font>)</span>)</span> &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H'</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">G'</font>) &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c1">X</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">H</font>) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="monoid_1.html#E1:43"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="monoid_1.html#T18" target="_self" title="MONOID_1:th.18">Th18</a>, <a class="ref" href="monoid_1.html#T23" target="_self" title="MONOID_1:th.23">Th23</a>, <a class="ref" href="monoid_0.html#T32" title="MONOID_0:th.32">MONOID_0:32</a>, <a class="ref" href="monoid_0.html#D23" title="MONOID_0:def.23">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" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c1">X</font>) <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K11" title="MONOID_1:func.11">.:</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">X</font>)
 
<b>by </b><i><a class="ref" href="monoid_1.html#T17" target="_self" title="MONOID_1:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E5:43"/>
 <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c1">X</font> is    <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of  <a href="monoid_1.html#K11" title="MONOID_1:func.11">.:</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E3:43"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="monoid_0.html#D25" title="MONOID_0:def.25">MONOID_0:def 25</a></i>;<br/>


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