<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:4"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:4_1"/><b>then </b>
( 0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; 0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="int_2.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:4"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


</div>
