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





<b>assume </b><a NAME="E1:4"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font> &amp; <font color="Maroon">b2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font> &amp; <font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><i><font color="Green">E45</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1'</font> = 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2'</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_2"/>
(  not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1'</font> or <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1"><b>suppose </b></a><a NAME="E1:4_1_2_1"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1'</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">b2'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:4_1_2_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> <font color="Maroon">b2'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_2"><b>suppose </b></a><a NAME="E1:4_1_2_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1'</font>
 ;<br/><div class="add"><a NAME="E2:4_1_2_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E3:4_1_2_2"/>
<font color="Maroon">b2'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:4_1_2_2"/>
<font color="Maroon">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2'</font> <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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
<font color="Maroon">b1'</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2'</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/>


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