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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span></span><br/>



<a NAME="E1:39"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span></span><br/>

<a NAME="E2:39_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> implies <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#T9" target="_self" title="NAT_D:th.9">Th9</a></i>;<br/>
<a NAME="E3:39_1"/>
( <font color="Maroon" title="c2">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c2">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">i</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#T9" target="_self" title="NAT_D:th.9">Th9</a></i>;<br/>
<a NAME="E4:39_1"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="txt" href="#E2:39_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="nat_d.html#D4" target="_self" title="NAT_D:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">c</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:39_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<a NAME="E7:39_1"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c2">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D4" target="_self" title="NAT_D:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:39_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:39_1"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="#E7:39_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<a NAME="E12:39_1"/>
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:39_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="#E9:39_1"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>
<a NAME="E13:39_1"/><b>then </b><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="txt" href="#E1:39_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E14:39_1"/>
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">e</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:39_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="#E11:39_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<a NAME="E15:39_1"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="#E1:39_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E16:39_1"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
( <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> )
 
<b>by </b><i><a class="txt" href="#E13:39_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<a NAME="E17:39_1"/>
for <font color="Olive" title="b1">f</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Olive" title="b1">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> holds <br/><font color="Olive" title="b1">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">f</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c6">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c6">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font> )</span><br/>

<b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green" title="E30">A10</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c6">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">g</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">h</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:39_1_1"/><i><font color="Green" title="E32">A12</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font>
 <b>by </b><i><a class="txt" href="#E1:39_1_1"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<a NAME="E6:39_1_1"/><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="#E5:39_1_1"><i><font color="Green" title="E32">A12</font></i></a></i>;<br/>
<a NAME="E7:39_1_1"/>
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/>
<a NAME="E8:39_1_1"/><b>then </b>
( <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c2">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font> )
 
<b>by </b><i><a class="txt" href="#E6:39_1_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<a NAME="E9:39_1_1"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D4" target="_self" title="NAT_D:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">z</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:39_1_1"/><i><font color="Green" title="E34">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">z</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<i><font color="Green" title="E35">A15</font></i>: <font color="Maroon" title="c3">c</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="#E6:39_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="#E5:39_1_1"><i><font color="Green" title="E32">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">z</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="#E11:39_1_1"><i><font color="Green" title="E34">A14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span>

;<br/>
<a NAME="E13:39_1_1"/>
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:39_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="int_2.html#T4" title="INT_2:th.4">INT_2:4</a></i>;<br/>
<a NAME="E14:39_1_1"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">z</font>
 
<b>by </b><i><a class="txt" href="#E12:39_1_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E15:39_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:39_1"/>
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:39_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="#E16:39_1"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:39"/>
( ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_2"/><i><font color="Green" title="E22">A16</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span></span><br/>

<b>then </b><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:39_2"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="int_2.html#T4" title="INT_2:th.4">INT_2:4</a></i>
;<br/>
<b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:39"/>
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E21">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
