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<span class="kw">let </span><font color="Maroon" title="c1">n1</font>, <font color="Maroon" title="c2">n2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="monoid_0.html#K8" title="MONOID_0:func.8">&lt;NAT,*,1&gt;</a>  st <font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m2</font> holds <br/><font color="Olive" title="b1">m1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c4">m2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="monoid_0.html#K8" title="MONOID_0:func.8">&lt;NAT,*,1&gt;</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m2</font> implies <font color="Maroon" title="c3">m1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n2</font> )</span><br/>



<a NAME="E1:119"/>
 <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 <a href="monoid_0.html#K8" title="MONOID_0:func.8">&lt;NAT,*,1&gt;</a>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K8" title="MONOID_0:func.8">&lt;NAT,*,1&gt;</a> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="monoid_0.html#K7" title="MONOID_0:func.7">&lt;NAT,*&gt;</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D22" target="_self" title="MONOID_0:def.22">Def22</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c4">m2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="monoid_0.html#K7" title="MONOID_0:func.7">&lt;NAT,*&gt;</a> ;<br/>
<a NAME="E3:119"/>
<font color="Maroon" title="c5">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T18" target="_self" title="MONOID_0:th.18">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:119"/>
( <font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m2</font> implies <font color="Maroon" title="c3">m1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T50" target="_self" title="MONOID_0:th.50">Th58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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