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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<b>assume </b><a NAME="E1:4_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">F<sub>1</sub></font>() <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#R1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 ;<br/>

<a NAME="E2:4_1"/><b>then </b><i><font color="Green">E3</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <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>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:4_1"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="polynom2.html#E2:4_1"><i><font color="Green">E3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>1</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom2.html#E1:4"><i><font color="Green">E1</font></i></a>, <a class="txt" href="polynom2.html#E4:4_1"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div class="add"><a NAME="E2:4_1_1_2"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom2.html#E4:4_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:4_1_1_2"/><b>then </b>
<font color="Maroon">F<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>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:4_1_1_2"/><b>then </b><i><font color="Green">E5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="polynom2.html#E4:4_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E6:4_1_1_2"/><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 0
 ;<br/><a NAME="E7:4_1_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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="E8:4_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E9:4_1_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom2.html#E7:4_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E10:4_1_1_2"/><b>then </b><i><font color="Green">E9</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="polynom2.html#E4:4_1"><i><font color="Green">E4</font></i></a></i>;<br/><a NAME="E11:4_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="polynom2.html#E4:4_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="polynom2.html#E9:4_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom2.html#E2:4"><i><font color="Green">E2</font></i></a>, <a class="txt" href="polynom2.html#E4:4_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="polynom2.html#E4:4_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="polynom2.html#E6:4_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="polynom2.html#E10:4_1_1_2"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
