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<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="int_1.html#NM1">Integer</a>;<br/>



<a NAME="E1:23"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K2" target="_self">lcm'</a> <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="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
0 <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K2" target="_self">lcm'</a> <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#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:23"/><i><font color="Green">E17</font></i>: 
( <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="int_2.html#K2" target="_self">lcm'</a> <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="23_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:23_2"/><b>then </b>
0 <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E3:23_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K2" target="_self">lcm'</a> <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#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:23"/>
(  not <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <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="23_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:23_3"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E3:23_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:23_3"/><b>then </b>
0 <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E1:23_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="int_2.html#E2:23_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="int_2.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E5:23_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <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#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E6:23_3"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:23"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <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="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="int_2.html#E1:23"><i><font color="Green">E16</font></i></a>, <a class="txt" href="int_2.html#E2:23"><i><font color="Green">E17</font></i></a></i>;<br/>


</div>
