<?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> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:28"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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>
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E19</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&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="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:28"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="polynom3.html#E1:28"><i><font color="Green">E18</font></i></a>, <a class="ref" href="binarith.html#T5">BINARITH:5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</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="E5:28"/><i><font color="Green">E21</font></i>: 
 <a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</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>3</sub></font>
 <b>and </b><br/><a NAME="E6:28"/><i><font color="Green">E22</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>3</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>1</sub></font> )
 <b>by </b><i><a class="ref" href="polynom3.html#D4" target="_self">Def4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<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"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> );<br/>
<a NAME="E7:28"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom3.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1"><a href="polynom3.html#K2" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:28"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</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>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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> )
 <b>and </b><br/><a NAME="E2:28_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:28_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <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="txt" href="polynom3.html#E2:28_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:28_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:28_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 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">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:28_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:28_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 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">c<sub>1</sub></font>,2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E7:28_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</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>1</sub></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="E8:28_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="polynom1.html#K3">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:28_2"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></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/>
<a NAME="E11:28_2"/>
 <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="E12:28_2"/><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="E13:28_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="polynom3.html#K5" target="_self">TuplesOrder</a> 2 <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E14:28_2"/>
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</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>1</sub></font>,2</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom3.html#E7:28_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:28_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E5:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom3.html#E13:28_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E16:28_2"/><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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E6:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom3.html#E10:28_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E17:28_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/>
<a NAME="E18:28_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E19:28_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E17:28_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E20:28_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E21:28_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E22:28_2"/><i><font color="Green">E35</font></i>: 
<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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="polynom3.html#E3:28_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E23:28_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E24:28_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="polynom3.html#E21:28_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E25:28_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:28_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E26:28_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E27:28_2"/>
<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="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:28_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green">E38</font></i>: <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="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E26:28_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<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="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom3.html#E22:28_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom3.html#E21:28_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom3.html#E24:28_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<b>then </b><font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="polynom3.html#E1:28_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom3.html#E2:28_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom3.html#E10:28_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom3.html#E19:28_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="polynom3.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E20:28_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/>
<b>hence </b><a NAME="E31:28_2"/>
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</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="txt" href="polynom3.html#E10:28_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom3.html#E19:28_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom3.html#E28:28_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:28"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="txt" href="polynom3.html#E7:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom3.html#E8:28"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:28"/>
<span class="p1">(<span class="default"><a href="polynom3.html#K6" target="_self">Decomp</a> <font color="Maroon">c<sub>1</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</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>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="polynom3.html#K2" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom3.html#E2:28"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom3.html#E3:28"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
