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<b>reconsider </b><font color="Maroon" title="c1">G</font> = <font color="Maroon">F<sub>1</sub></font>() as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  ;<br/>
<a NAME="E2:98_1"/><i><font color="Green" title="E49">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font>]
 
<b>by </b><i><a class="txt" href="monoid_0.html#E1:98"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/>
<a NAME="E3:98_1"/><i><font color="Green" title="E50">A4</font></i>: 
 ex <font color="Olive" title="b1">x</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>]
 
<b>by </b><i><a class="txt" href="monoid_0.html#E2:98"><i><font color="Green" title="E48">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">H</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:98_1"/><i><font color="Green" title="E51">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 <b>from </b><i><a class="ref" href="monoid_0.html#S2" target="_self" title="MONOID_0:sch.2">MONOID_0:sch 2</a>(<a class="txt" href="monoid_0.html#E2:98_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="monoid_0.html#E3:98_1"><i><font color="Green" title="E50">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">e</font> =  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon">F<sub>1</sub></font>() as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="monoid_0.html#E2:98"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="monoid_0.html#E5:98_1"><i><font color="Green" title="E51">A5</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">M</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">H</font>),<font color="Maroon" title="c3">e</font> #);<br/>
<a NAME="E7:98_1"/><i><font color="Green" title="E52">A6</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">H</font>),<font color="Maroon" title="c3">e</font> #)
 
;<br/>
<a NAME="E8:98_1"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c2">H</font> <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">Def23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">M</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">H</font>),<font color="Maroon" title="c3">e</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">strict</a>  <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="monoid_0.html#E7:98_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="monoid_0.html#D25" target="_self" title="MONOID_0:def.25">Def25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">M</font>
;<br/>

<b>thus </b><a NAME="E10:98_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">M</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 <b>by </b><i><a class="txt" href="monoid_0.html#E5:98_1"><i><font color="Green" title="E51">A5</font></i></a></i>;<br/>


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