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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></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="E2:26"/><i><font color="Green">E16</font></i>: 
 <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff  <a href="wsierp_1.html#K8">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polynom3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:26"/>
 <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="E5:26"/><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="E6:26"/><b>then </b>
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> 2 <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E7:26"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:26"><i><font color="Green">E16</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E8:26"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>and </b><br/><a NAME="E9:26"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 ;<br/>

<a NAME="E10:26"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:26"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E8:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom3.html#E9:26"><i><font color="Green">E20</font></i></a>, <a class="ref" href="polynom3.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E12:26"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>assume </b><a NAME="E13:26"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E14:26"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E12:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E15:26"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:26_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E11:26"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:26_1"/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/>
<a NAME="E4:26_1"/><b>then </b>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:26_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom3.html#E3:26"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom3.html#E7:26"><i><font color="Green">E18</font></i></a>, <a class="txt" href="polynom3.html#E9:26"><i><font color="Green">E20</font></i></a>, <a class="txt" href="polynom3.html#E2:26_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E6:26_1"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 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">c<sub>2</sub></font>,2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:26_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom3.html#E9:26"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:26"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E16:26"><i><font color="Green">E24</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/>
<a NAME="E18:26"/><b>then </b>
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom3.html#E3:26"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom3.html#E7:26"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E20:26"/>
<font color="Maroon">c<sub>6</sub></font> <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">c<sub>2</sub></font>,2</span>)</span>
 <b>and </b><br/><a NAME="E21:26"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E22:26"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E23:26"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E8:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom3.html#E21:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom3.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E24:26"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E25:26"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom3.html#E9:26"><i><font color="Green">E20</font></i></a>, <a class="txt" href="polynom3.html#E14:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom3.html#E21:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom3.html#T11" target="_self">Th11</a></i>;<br/>


</div>
