<?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> ;<br/><b>let </b><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>;<br/>



<div><i><font color="Green" title="E7">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">b1</font>,<font color="Maroon" title="c3">b2</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_1_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</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="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="groeb_2.html#E1:11_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="groeb_2.html#D3" target="_self" title="GROEB_2:def.3">Def3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1_1"><b>case </b></a><a NAME="E1:11_1_1_1_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:11_1_1_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1_2"><b>case </b></a><a NAME="E1:11_1_1_1_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:11_1_1_1_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:11_1"/>
 <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c2">b1</font>,<font color="Maroon" title="c3">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#D2" target="_self" title="GROEB_2:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2"/><i><font color="Green" title="E8">A5</font></i>: 
 <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c2">b1</font>,<font color="Maroon" title="c3">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:11_2_1"/><i><font color="Green" title="E9">A6</font></i>: 
<span class="p1">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c2">b1</font>,<font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> )
 </span><b>by </b><i><a class="txt" href="groeb_2.html#E1:11_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="groeb_2.html#E1:11_2_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_1_1"><b>case </b></a><a NAME="E1:11_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font>
 ;<br/><div class="add"><a NAME="E2:11_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:11_2_1_1_1_1"/>
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_1_2"><b>case </b></a><a NAME="E1:11_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font>
 ;<br/><div class="add"><a NAME="E2:11_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:11_2_1_1_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:11_2_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</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="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:11_2"/>
<font color="Maroon" title="c2">b1</font>,<font color="Maroon" title="c3">b2</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a> 
 <b>by </b><i><a class="ref" href="groeb_2.html#D3" target="_self" title="GROEB_2:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11"/>
( <font color="Maroon" title="c2">b1</font>,<font color="Maroon" title="c3">b2</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  iff  <a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <font color="Maroon" title="c2">b1</font>,<font color="Maroon" title="c3">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c3">b2</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:11"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>


</div>
