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<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="E1:18"/>
 <a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K6">lcm</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K6">lcm</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 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="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K6">lcm</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E4:18_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="int_2.html#E1:18_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:18_1"/>
<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="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_1.html#D9">INT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:18"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 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="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K6">lcm</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="int_2.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="int_2.html#E1:18_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:18_2"/>
<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="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_1.html#D9">INT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:18"/>
<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="int_2.html#K2" target="_self">lcm'</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="int_2.html#E4:18"><i><font color="Green">E14</font></i></a></i>;<br/>


</div>
