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

<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="E2:11_1"/><i><font color="Green">E11</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:11"><i><font color="Green">E10</font></i></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">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E3:11_1"/><i><font color="Green">E12</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>]
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E2:11_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</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>2</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="fdiff_2.html#E3:11_1"><i><font color="Green">E12</font></i></a>);<br/></i>
<div><i><font color="Green">E14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<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="E2:11_1_1"/><i><font color="Green">E15</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> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>] )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:11"><i><font color="Green">E10</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E3:11_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>] )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E2:11_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>] &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> ) );<br/>
<a NAME="E7:11_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:11_1_2"/><i><font color="Green">E16</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>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E6:11_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:11_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>5</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>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>5</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="fdiff_2.html#E1:11_1_2"><i><font color="Green">E16</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E4:11_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:11_1_2"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:11_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="fdiff_2.html#E7:11_1"><i><font color="Green">E15</font></i></a>);<br/></i>
<a NAME="E11:11_1"/><i><font color="Green">E17</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:11_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="fdiff_2.html#E11:11_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:11_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E11:11_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:11_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E11:11_1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:11_1_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="nat_1.html#NM1">Nat</a> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E14:11_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>hence </b><a NAME="E2:11_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E10:11_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i><a class="txt" href="fdiff_2.html#E11:11_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>1</sub></font>() <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E17:11_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:11_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_2.html#E10:11_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/>
<a NAME="E18:11_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</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="11_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>]
 ;<br/>

<a NAME="E2:11_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E10:11_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E3:11_1_5"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>]
 
;<br/>
<b>hence </b><a NAME="E4:11_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E19:11_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fdiff_2.html#E17:11_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fdiff_2.html#E18:11_1"><i><font color="Green">E20</font></i></a>);</i><br/>

<a NAME="E20:11_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>] holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<b>assume </b><a NAME="E1:11_1_6"/><i><font color="Green">E22</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>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:11_1_6"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>8</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>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>8</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="fdiff_2.html#E1:11_1_6"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>5</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>8</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>] &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:11_1_6"/><i><font color="Green">E24</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>5</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<a NAME="E1:11_1_6_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:11_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fdiff_2.html#E3:11_1_6"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E2:11_1_6_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E10:11_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fdiff_2.html#E3:11_1_6"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:11_1_6_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:11_1_6_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:11_1_6_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="fdiff_2.html#E5:11_1"><i><font color="Green">E13</font></i></a></i>;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E5:11_1_6_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E10:11_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:11_1_6"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:11_1_6"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>9</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>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="fdiff_2.html#E5:11_1_6"><i><font color="Green">E25</font></i></a>, <a class="txt" href="fdiff_2.html#E4:11_1_6"><i><font color="Green">E24</font></i></a>);<br/></i>
<a NAME="E8:11_1_6"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>9</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">c<sub>8</sub></font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:11_1_6_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>] )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_6_2_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_2_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_6_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_6_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_2.html#E7:11_1_6"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_2.html#E1:11_1_6_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_2_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_6_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_6_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:11_1_6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fdiff_2.html#E1:11_1_6_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1_6_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:11_1_6"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E7:11_1_6"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E11:11_1_6"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E10:11_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fdiff_2.html#E10:11_1_6"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E12:11_1_6"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E10:11_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fdiff_2.html#E3:11_1_6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fdiff_2.html#E7:11_1_6"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fdiff_2.html#E10:11_1_6"><i><font color="Green">E28</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_6_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:11_1_6_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E12:11_1_6"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:11_1_6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_2.html#E8:11_1_6"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fdiff_2.html#E11:11_1_6"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:11_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:11_1_6"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E21:11_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<a NAME="E22:11_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>]
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E24:11_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E20:11_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E25:11_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E24:11_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div>
