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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">2 <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="E2:27"/><i><font color="Green">E33</font></i>: 
 <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">n</font>,2 <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> 2</span>)</span>,<font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of 2 <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="E4:27"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:27"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">n</font>,2</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:27"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">n</font>,2</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:27"/>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/>
<a NAME="E9:27"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:27"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 2 <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="E11:27"/><b>then </b>
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> 2 <a href="orders_1.html#R3">linearly_orders</a> 2 <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="E12:27"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> 2 <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">d2</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="E2:27_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120">FINSEQ_2:120</a></i>;<br/><b>assume </b><a NAME="E3:27_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E4:27_1"/><b>then </b>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:27_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">d1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1"/>
( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">d1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1"/>
<font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_1"/>
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="polynom3.html#R2" target="_self">&lt;=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2"/>
<font color="Maroon">d1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:27_1_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E4:27_1_1_1_2"/><i><font color="Green">E65</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:27_1_1_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/><span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 ;<br/><a NAME="E6:27_1_1_1_2"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="polynom3.html#R1" target="_self">&lt;</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:27_1_1_1_2"/>
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="polynom3.html#R2" target="_self">&lt;=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:27_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom3.html#K5" target="_self">TuplesOrder</a> 2
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E14:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom3.html#K5" target="_self">TuplesOrder</a> 2</span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="polynom1.html#K3">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#T8">POLYNOM1:8</a></i>;<br/>
<b>hence </b><a NAME="E15:27"/>
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">n</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">n</font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


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