<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">p1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">A1</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> </span>)</span><b> such that </b><a NAME="E1:20_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">A1</font>
 <b>and </b><br/><a NAME="E2:20_1_2"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> iff  <a href="wsierp_1.html#K7">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>

<b>given </b><font color="Maroon">A2</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> </span>)</span><b> such that </b><a NAME="E3:20_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">A2</font>
 <b>and </b><br/><a NAME="E4:20_1_2"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> iff  <a href="wsierp_1.html#K7">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:20_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:20_1_2_1_1"/>
 <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:20_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:20_1_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E4:20_1_2_1"/>
 <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:20_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:20_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
