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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>consider </b><font color="Maroon">A</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">1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E2:22"/><i><font color="Green">E33</font></i>: 
 <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">n</font>,1 <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> 1</span>)</span>,<font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of 1 <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">A</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> )
 <b>by </b><i/>;<br/>
<a NAME="E4:22"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:22_1_1"/><i><font color="Green">E44</font></i>: 
 <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>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:22_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T117">FINSEQ_2:117</a></i>;<br/>
<a NAME="E6:22_1_1"/>
 <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>;<br/>
<b>hence </b><a NAME="E7:22_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:22_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:22_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:22_1"/>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> 1 <a href="orders_1.html#R3">linearly_orders</a> 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/>
<a NAME="E7:22"/><b>then </b>
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> 1 <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<b>hence </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">n</font>,1</span>)</span> = 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="triang_1.html#T9">TRIANG_1:9</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/><br/>


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