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<b>let </b><font color="Maroon" title="c1">b</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  iff <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K7" title="MONOID_0:func.7">&lt;NAT,*&gt;</a>  )</span><br/>

<a NAME="E1:117_1_1"/><i><font color="Green" title="E53">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="monoid_0.html#K7" title="MONOID_0:func.7">&lt;NAT,*&gt;</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="monoid_0.html#D31" target="_self" title="MONOID_0:def.31">Def31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">n1</font>, <font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K7" title="MONOID_0:func.7">&lt;NAT,*&gt;</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font></span><br/><b>thus </b><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font> = 
<font color="Maroon" title="c2">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n2</font>
<b>by </b><i><a class="ref" href="binop_2.html#D24" title="BINOP_2:def.24">BINOP_2:def 24</a></i>
<br/>.= 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font>
<b>by </b><i><a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K7" title="MONOID_0:func.7">&lt;NAT,*&gt;</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:117_1_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="monoid_0.html#T26" target="_self" title="MONOID_0:th.26">Th26</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>
<b>hence </b><a NAME="E3:117_1_1"/>
( <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  iff <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K7" title="MONOID_0:func.7">&lt;NAT,*&gt;</a>  )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:117_1_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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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