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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> ; <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">H1</font>, <font color="Olive" title="b2">H2</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of <font color="Maroon" title="c1">M</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> holds <br/> <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="Olive" title="b1">H1</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">H1</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Olive" title="b1">H1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.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="Olive" title="b2">H2</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b2">H2</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Olive" title="b2">H2</font> #)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of <font color="Maroon" title="c1">M</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="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font> implies  <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="c2">H1</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">H1</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c2">H1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.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">H2</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H2</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c3">H2</font> #) )</span><br/>



<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="c2">H1</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">H1</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c2">H1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.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">H2</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H2</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c3">H2</font> #)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">N1</font> = <font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c5">N2</font> = <font color="Maroon" title="c3">H2</font> as    <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T23" target="_self" title="MONOID_0:th.23">Th23</a></span>;<br/>
<a NAME="E3:77"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">H1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c3">H2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</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/>
<a NAME="E4:77"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">N1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">N2</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c5">N2</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="monoid_0.html#T28" target="_self" title="MONOID_0:th.28">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:77"/>
 <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="c2">H1</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">H1</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c2">H1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.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">H2</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H2</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c3">H2</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E40">A2</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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