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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c2">n</font>;<br/>
<b>set </b><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span>;<br/>
<a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
( <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E2:1"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T8" title="NAT_D:th.8">NAT_D:8</a></i>;<br/>
<a NAME="E3:1"/><b>then </b><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E4:1"/><i><font color="Green" title="E4">A3</font></i>: 
( <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E5:1"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T10" title="NAT_D:th.10">NAT_D:10</a></i>;<br/>
<a NAME="E6:1"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E4:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:1"/>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="fib_num.html#E3:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="nat_d.html#T5" title="NAT_D:th.5">NAT_D:5</a></i>;<br/>


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