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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">k1</font>, <font color="Maroon" title="c5">k2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon" title="c6">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E2:25"/><i><font color="Green" title="E15">A1</font></i>: 
 <a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2</span>)</span>,<font color="Maroon" title="c6">A</font>
 <b>and </b><br/><a NAME="E3:25"/>
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> iff  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 <b>by </b><i><a class="ref" href="polynom3.html#D4" target="_self" title="POLYNOM3:def.4">Def4</a></i>;<br/>
<a NAME="E4:25"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<a NAME="E5:25"/><b>then </b>
 <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2 <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a></i>;<br/>
<a NAME="E6:25"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2 <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E7:25"/><i><font color="Green" title="E17">A3</font></i>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E8:25"/><i><font color="Green" title="E18">A4</font></i>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 ;<br/>

<a NAME="E9:25"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> )
 
<b>by </b><i><a class="txt" href="polynom3.html#E7:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="polynom3.html#E8:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E10:25"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c1">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c3">n</font>,2</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c2">j</font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>thus </b><a NAME="E11:25"/>
( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
( <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>,<span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2 )
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="polynom3.html#E6:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="polynom3.html#E7:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="polynom3.html#E8:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="polynom3.html#E9:25"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="polynom3.html#E10:25"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E3:25_1"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="polynom3.html#D3" target="_self" title="POLYNOM3:def.3">Def3</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:25_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:25_1"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>and </b><br/><a NAME="E7:25_1"/><i><font color="Green" title="E23">A9</font></i>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c7">t</font>
 <b>and </b><br/><a NAME="E8:25_1"/><i><font color="Green" title="E24">A10</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">t</font> holds <br/><span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="polynom3.html#E6:25_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:25_1_1_1"/><b>then </b>
( <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k1</font> &amp; <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1"/>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="txt" href="polynom3.html#E7:25_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:25_1_1_2"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> 1
 <b>by </b><i><a class="txt" href="polynom3.html#E8:25_1"><i><font color="Green" title="E24">A10</font></i></a></i>;<br/><a NAME="E3:25_1_1_2"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E4:25_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_2"/>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="txt" href="polynom3.html#E7:25_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>assume </b><a NAME="E12:25"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font>
 ;<br/>

<a NAME="E13:25"/><i><font color="Green" title="E22">A12</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:25"/><i><font color="Green" title="E23">A13</font></i>: 
( <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k1</font> &amp; <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E15:25"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font>
 
;<br/>
<a NAME="E16:25"/><b>then </b><i><font color="Green" title="E24">A14</font></i>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom3.html#E12:25"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="polynom3.html#E13:25"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="polynom3.html#E14:25"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E17:25"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> )
 </span><b>by </b><i><a class="txt" href="polynom3.html#E17:25"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><b>suppose </b></a><a NAME="E1:25_2_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom3.html#E7:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="polynom3.html#E8:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="polynom3.html#E12:25"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="polynom3.html#E14:25"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2"><b>suppose </b></a><a NAME="E1:25_2_2"/>
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font>
 ;<br/><div class="add"><a NAME="E2:25_2_2"/><b>then </b><i><font color="Green" title="E26">A16</font></i>: 
( <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>,<span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2 )
 <b>by </b><i><a class="txt" href="polynom3.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="polynom3.html#E6:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="polynom3.html#E7:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="polynom3.html#E8:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="polynom3.html#E9:25"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="polynom3.html#E10:25"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E3:25_2_2"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom3.html#D3" target="_self" title="POLYNOM3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:25_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom3.html#E16:25"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="polynom3.html#E2:25_2_2"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
