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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<a NAME="E1:63"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K6">lcm</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K6">lcm</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K6">lcm</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:63_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K6">lcm</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K6">lcm</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K6">lcm</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="newton.html#E1:63_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:63"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K6">lcm</a> <font color="Maroon">c<sub>3</sub></font> <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#D4">NAT_1:def 4</a></i>;<br/>


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