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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#V3" title="STRUCT_0:attr.3">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c2">D</font> = <font color="Maroon">F<sub>2</sub></font>() as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E3:99_1"/><i><font color="Green" title="E50">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">D</font> holds  <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E1:99"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">H</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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="E5:99_1"/><i><font color="Green" title="E51">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font>
 <b>from </b><i><a class="ref" href="monoid_0.html#S1" target="_self" title="MONOID_0:sch.1">MONOID_0:sch 1</a>(<a class="txt" href="monoid_0.html#E3:99_1"><i><font color="Green" title="E50">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">e</font> = the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <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="c3">H</font> <b>by </b><i><a class="txt" href="monoid_0.html#E2:99"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="monoid_0.html#E5:99_1"><i><font color="Green" title="E51">A4</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">N</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">e</font> #);<br/>
<a NAME="E7:99_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">e</font> #) is    <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:99_1_1"/>
<font color="Maroon">H<sub>2</sub></font>( <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">e</font> #)) <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>(<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D25" target="_self" title="MONOID_0:def.25">MONOID_0:def 25</a><br/>

<b>thus </b><a NAME="E2:99_1_1"/>
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">e</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">N</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">e</font> #) as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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>() ;<br/>
<b>take </b>
<font color="Maroon" title="c6">N</font>
;<br/>

<b>thus </b><a NAME="E9:99_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="monoid_0.html#E5:99_1"><i><font color="Green" title="E51">A4</font></i></a></i>;<br/>


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