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<a NAME="E1:143_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">b</font> as  <a href="polynom1.html#V1" target="_self">finite-support</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>, <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> =  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#T10" target="_self">Th10</a></i>;<br/>
<b>take </b><font color="Maroon">IT</font> =  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font>;<br/>

<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E4:143_1_1"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>

<b>thus </b><a NAME="E5:143_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">p</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:143_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:143_1_1_1"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:143_1_1_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E6:143_1_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E7:143_1_1"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">p</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:143_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T9" target="_self">Th9</a></i>;<br/>


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