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<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>, <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>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E9">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>
 ;<br/>

<a NAME="E2:14"/>
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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_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> ;<br/>

<b>assume </b><a NAME="E1:14_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>
 ;<br/>

<a NAME="E2:14_1"/>
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</a></i>;<br/>
<a NAME="E3:14_1"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="E4:14_1"/>
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="E5:14_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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E3:14_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="square_1.html#T50" title="SQUARE_1:th.50">SQUARE_1:50</a></i>;<br/>
<a NAME="E6:14_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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="E7:14_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="polynom1.html#K8" title="POLYNOM1:func.8">.</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="polynom1.html#K8" title="POLYNOM1:func.8">.</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
 <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>;<br/>


</div>
