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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="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</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="E3:96_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="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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#E1:96"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">H</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">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="E5:96_1"/><i><font color="Green" title="E50">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#E3:96_1"><i><font color="Green" title="E49">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">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="c3">H</font> <b>by </b><i><a class="txt" href="monoid_0.html#E2:96"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="monoid_0.html#E5:96_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">N</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">e</font> #);<br/>
<a NAME="E7:96_1"/>
 <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="96_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:96_1_1"/>
<font color="Maroon">H<sub>2</sub></font>( <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="E2:96_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <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="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">e</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V22" title="ALGSTR_0:attr.22">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="E9:96_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#E5:96_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>


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