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<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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="E3:98_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:98_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
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>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="monoid_0.html#S1" target="_self" title="MONOID_0:sch.1">MONOID_0:sch 1</a>(<a class="txt" href="#E3:98_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E5:98_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a></span>;<br/>
<span class="kw">set </span><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="E7:98_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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">F<sub>1</sub></font>()) &amp; ( for <font color="Olive" title="b1">M</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>   st <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> holds <br/> <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="Olive" title="b1">M</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">Def23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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>() <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D25" target="_self" title="MONOID_0:def.25">Def25</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">N</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>()</span><br/>

<span class="kw">thus </span><a NAME="E9:98_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>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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