<?xml version="1.0"?>
<div class="add">

<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="E1:5"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:5_1"/><i><font color="Green">E12</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="E3:5_1"/>
( <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="E4:5_1"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</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#D4">NAT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:5_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E7:5_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">a</font> <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> &amp; <font color="Maroon">b</font> <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>then consider </b><font color="Maroon">d</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:5_1"/><i><font color="Green">E20</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> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:5_1"/><i><font color="Green">E22</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> <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E12:5_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:5_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E14:5_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:5_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E16:5_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="txt" href="int_2.html#E1:5"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E17:5_1"/>
for <font color="Olive">f</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">f</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> &amp; <font color="Olive">f</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> holds <br/><font color="Olive">f</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">f</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:5_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:5_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="int_2.html#E2:5_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E6:5_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:5_1_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:5_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E9:5_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font>
 
<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">z</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:5_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<i><font color="Green">E34</font></i>: <font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font>
<b>by </b><i>, , <a class="ref" href="int_2.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="int_2.html#E6:5_1"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span>

;<br/>
<a NAME="E13:5_1_1"/>
<font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:5_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E15:5_1_1"/>
<font color="Maroon">f</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:5_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="int_2.html#E1:5_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:5"/>
( ( <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 ) implies <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E35</font></i>: 
( <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 )
 ;<br/>

<b>then </b><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> = 
0 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:5_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
