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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>thus </b><a NAME="E1:31"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> implies ( <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:31_1"/>
 ex <font color="Olive">q</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">q</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:31_1"/>
( <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 ;<br/>


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

<b>thus </b><a NAME="E2:31"/>
( <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> )
 <b>by </b><i/>;<br/>


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