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<b>consider </b><font color="Maroon" title="c2">M</font> being    <a href="monoid_0.html#M1" title="MONOID_0:mode.1">MonoidalExtension</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E2:60_1_1"/><i><font color="Green" title="E31">A1</font></i>: 
 <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="c2">M</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">M</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="c1">G</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> #)
 
<b>by </b><i><a class="ref" href="monoid_0.html#D22" target="_self" title="MONOID_0:def.22">Def22</a></i>;<br/>
<div><i><font color="Green" title="E32">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c3">M</font> being    <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> <b> such that </b><a NAME="E2:60_1_1_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">M</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>  st <br/>( <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">M</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; ( for <font color="Olive" title="b2">N</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>   st <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">N</font>) ) )</span><br/><b>take </b><font color="Maroon" title="c4">M</font> = <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">M</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; ( for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>   st <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c4">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">N</font>) ) )</span><br/><b>thus </b><a NAME="E3:60_1_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">M</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:60_1_1_1"><i><font color="Green" title="E33">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>   st <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c4">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">N</font>)</span><br/><b>thus </b><a NAME="E4:60_1_1_1"/>
for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>   st <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c4">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">N</font>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:60_1_1_1"><i><font color="Green" title="E33">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:60_1_1"/>
(  ex <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" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> or for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>   st <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">N</font>) )
 
;<br/>
<b>hence </b><a NAME="E5:60_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>  st <br/>( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive">b<sub>1</sub></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" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">M</font> being   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a>   st <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> holds <br/> <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive">b<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> <font color="Olive" title="b2">M</font> ) )
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:60_1_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="monoid_0.html#E3:60_1_1"><i><font color="Green" title="E32">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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