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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E10</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E11</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="polynom3.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <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="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E1:17"><i><font color="Green">E10</font></i></a>, <a class="ref" href="polynom3.html#T6" target="_self">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="polynom3.html#D1" target="_self">POLYNOM3:def 1</a><br/>

<b>thus </b><a NAME="E7:17"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E4:17"><i><font color="Green">E12</font></i></a></i>;<br/>

<a NAME="E8:17"/>
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="polynom3.html#E4:17"><i><font color="Green">E12</font></i></a>, <a class="txt" href="polynom3.html#E6:17"><i><font color="Green">E14</font></i></a>, <a class="ref" href="polynom3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E9:17"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E5:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E10:17"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E6:17"><i><font color="Green">E14</font></i></a></i>;<br/>


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