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<div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>set </b><font color="Maroon">M</font> = <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">K</font> = <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">N</font> = <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>;<br/>
<a NAME="E1:61"/>
<font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">l</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">l</font> holds <br/><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">l</font> ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<a NAME="E2:61_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">i</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">i</font> holds <br/><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Olive">i</font> ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<a NAME="E3:61_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">s</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">s</font> holds <br/><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">s</font> ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<a NAME="E4:61_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">j</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">j</font> ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<a NAME="E5:61_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="newton.html#E2:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="newton.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E2:61_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="newton.html#E2:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="newton.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E3:61_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="newton.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="newton.html#E3:61_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:61_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T3" target="_self">Th3</a>, <a class="txt" href="newton.html#E1:61_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E2:61_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T3" target="_self">Th3</a>, <a class="txt" href="newton.html#E1:61_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E3:61_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E1:61_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="newton.html#E3:61_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:61_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T52">NAT_1:52</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:61"/>
<font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 ;<br/>


</div>
