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<div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<a NAME="E1:24"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:24_1"/><b>then </b>
0 <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1"/>
<font color="Maroon">a</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:24"/>
( <font color="Maroon">a</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 implies ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_2"/>
<font color="Maroon">a</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:24_2"/><b>then </b>
( 0 <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> &amp; 0 <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E3:24_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) iff <font color="Maroon">a</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>


</div>
