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<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F<sub>1</sub></font>() as  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a>  ;<br/>
<a NAME="E2:100_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:100_1"/><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">H</font> being  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:100_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] )
 <b>from </b><i><a class="ref" href="monoid_0.html#S2" target="_self">MONOID_0:sch 2</a>(<a class="ref" href="monoid_0.html#D12" target="_self">Def12</a>, <a class="ref" href="monoid_0.html#D16" target="_self">Def16</a>);<br/></i>
<b>reconsider </b><font color="Maroon">e</font> = the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">F<sub>1</sub></font>() as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i>, <a class="ref" href="monoid_0.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E7:100_1"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> =  <a href="vectsp_1.html#G1">multLoopStr</a>(# <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H</font>),<font color="Maroon">e</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V1">strict</a>  <a href="monoid_0.html#M3" target="_self">MonoidalSubStr</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E9:100_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] )
 <b>by </b><i><a class="ref" href="monoid_0.html#D20" target="_self">Def20</a></i>;<br/>


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