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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">X</font>, <font color="Maroon">b2</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">X</font>, <font color="Maroon">b3</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green">E31</font></i>: 
 <a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>
 ;<br/>

<a NAME="E2:15"/>
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b1</font>,<font color="Maroon">b3</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:15_1"/>
<span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/>
<a NAME="E3:15_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:15_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:15_1"/>
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:15_1"/>
<font color="Maroon">b3</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E8:15_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:15_1"/>
<span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b1</font>,<font color="Maroon">b3</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
 <a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b1</font>,<font color="Maroon">b3</font> <a href="polynom1.html#R3">divides</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <font color="Maroon">b2</font>,<font color="Maroon">b3</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T50">POLYNOM1:50</a></i>;<br/>


</div>
