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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="monoid_1.html#NM2" title="MONOID_1:NM.2">Multiset</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">m1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">m2</font></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m1</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m2</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Olive" title="b3">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">m1</font>, <font color="Maroon" title="c3">m2</font> be   <a href="monoid_1.html#NM2" title="MONOID_1:NM.2">Multiset</a> of <font color="Maroon" title="c1">D</font>; <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">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">a</font></span>)</span></span><br/>





<b>reconsider </b><font color="Maroon" title="c5">N</font> =  <a href="monoid_0.html#K5" title="MONOID_0:func.5">&lt;NAT,+,0&gt;</a>  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="c6">f1</font> = <font color="Maroon" title="c2">m1</font>, <font color="Maroon" title="c7">f2</font> = <font color="Maroon" title="c3">m2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">D</font></span>)</span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="monoid_1.html#K10" title="MONOID_1:func.10">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="monoid_1.html#K10" title="MONOID_1:func.10">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_1.html#T22" target="_self" title="MONOID_1:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_0.html#T51" title="MONOID_0:th.51">MONOID_0:51</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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