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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/>
<a NAME="E2:99_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 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>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>;<br/>
<a NAME="E3:99_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">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="E5:99_1"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
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>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="monoid_0.html#S2" target="_self" title="MONOID_0:sch.2">MONOID_0:sch 2</a>(<a class="txt" href="#E3:99_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E2:99_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E5:99_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">M</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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="E7:99_1"/>
(  <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="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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> #) &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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#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="c5">M</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">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="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="c5">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>] )</span><br/>

<span class="kw">thus </span><a NAME="E9:99_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>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_1"><span class="lab"><font color="Green" title="E52">A5</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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