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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="nat_lat.html#M1" target="_self">Element</a> of  <a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,   <a href="nat_lat.html#M1" target="_self">Element</a> of  <a href="nat_lat.html#K7" target="_self">NATPLUS</a> ) <b>-&gt; </b>   <a href="nat_lat.html#M1" target="_self">Element</a> of  <a href="nat_lat.html#K7" target="_self">NATPLUS</a>  =  <a href="nat_lat.html#K8" target="_self">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p2">(<span class="default"><a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="nat_lat.html#K7" target="_self">NATPLUS</a> <b> such that </b><br/><a NAME="E2:32_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="nat_lat.html#M1" target="_self">Element</a> of  <a href="nat_lat.html#K7" target="_self">NATPLUS</a>  holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">m</font>,<font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>,<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="nat_lat.html#M1" target="_self">Element</a> of  <a href="nat_lat.html#K7" target="_self">NATPLUS</a> , <font color="Maroon">n</font> be    <a href="nat_lat.html#M1" target="_self">Element</a> of  <a href="nat_lat.html#K7" target="_self">NATPLUS</a> ;<br/>

<a NAME="E3:32_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:32_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="nat_lat.html#K9" target="_self">@</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">n</font> )
 
;<br/>
<a NAME="E5:32_1_1"/>
(  <a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="nat_lat.html#K9" target="_self">@</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="nat_lat.html#K9" target="_self">@</a> <font color="Maroon">m</font> )
 
;<br/>
<a NAME="E6:32_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="nat_lat.html#K8" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:32_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="newton.html#T71">NEWTON:71</a></i>;<br/>
<b>hence </b><a NAME="E8:32_1_1"/>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_lat.html#D3" target="_self">Def3</a>, </i>;<br/>


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