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<b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:3"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:3_1"/><b>then </b>
0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</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="E3:3_1"/>
<font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</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="E2:3"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:3_2"/><b>then </b>
0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</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="E3:3_2"/>
<font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</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="E3:3"/>
(  not <font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:3_3"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> implies <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<a NAME="E3:3_3"/>
( <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> implies <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<a NAME="E4:3_3"/><b>then </b>
0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#D4">NAT_1:def 4</a></i>;<br/>
<a NAME="E5:3_3"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:3_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</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="E4:3"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) iff <font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, </i>;<br/>


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