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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font>, <font color="Olive" title="b3">b3</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Olive" title="b2">b2</font>,<font color="Olive" title="b3">b3</font> holds <br/> <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Olive" title="b2">b2</font>,<font color="Olive" title="b1">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Olive" title="b2">b2</font>,<font color="Olive" title="b3">b3</font></span><br/><b>let </b><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font>, <font color="Maroon" title="c4">b3</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font> implies  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font> )</span><br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font></span><br/>

<a NAME="E2:13"/>
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c2">b1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c2">b1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> )</span><br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c2">b1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span><br/>

<a NAME="E2:13_1"/>
<font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</a></i>;<br/>
<a NAME="E3:13_1"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_2.html#D2" target="_self" title="GROEB_2:def.2">Def2</a></i>;<br/>
<a NAME="E4:13_1"/>
<font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E5:13_1"/><b>then </b>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b3</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:13_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T28" title="XXREAL_0:th.28">XXREAL_0:28</a></i>;<br/>
<a NAME="E6:13_1"/><b>then </b>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="ref" href="groeb_2.html#D2" target="_self" title="GROEB_2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1"/>
<span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c2">b1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#D2" target="_self" title="GROEB_2:def.2">Def2</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="E3:13"/>
 <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c3">b2</font>,<font color="Maroon" title="c4">b3</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T50" title="POLYNOM1:th.50">POLYNOM1:50</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>
