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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:110_1_1"/><i><font color="Green">E52</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="monoid_0.html#K5" target="_self">&lt;NAT,+&gt;</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="monoid_0.html#D27" target="_self">Def27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:110_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/><b>thus </b><a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D23">BINOP_2:def 23</a></i>
<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:110_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="binop_2.html#D20">BINOP_2:def 20</a>, <a class="ref" href="numbers.html#T17">NUMBERS:17</a></i>
<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="monoid_0.html#K4" target="_self">INT.Group</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D4">GR_CY_1:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="monoid_0.html#K5" target="_self">&lt;NAT,+&gt;</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:110_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="monoid_0.html#T26" target="_self">Th26</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:110_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K47">addnat</a>  iff <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="monoid_0.html#K5" target="_self">&lt;NAT,+&gt;</a>  )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:110_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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