<?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="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E3:1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E6:1"/><i><font color="Green">E6</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E2:1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E7:1"/><i><font color="Green">E7</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E8:1"/><b>then </b><i><font color="Green">E8</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:1"/><i><font color="Green">E9</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="rcomp_1.html#V1">compact</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T24">RCOMP_1:24</a></i>;<br/>
<a NAME="E10:1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rolle.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</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="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E12:1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="rolle.html#E2:1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E9:1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="fcont_1.html#T32">FCONT_1:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E14:1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<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">c<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rolle.html#E1:1"><i><font color="Green">E2</font></i></a></i>;<br/>
<a NAME="E15:1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E16:1"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E17:1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="rcomp_1.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="rolle.html#E2:1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E9:1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="fcont_1.html#T30">FCONT_1:30</a></i>;<br/>
<a NAME="E18:1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T28">RCOMP_1:28</a></i>;<br/>
<a NAME="E19:1"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="seq_4.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E20:1"/><i><font color="Green">E14</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E16:1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rolle.html#E18:1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> )
 </span><b>by </b><i><a class="txt" href="rolle.html#E20:1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="partfun2.html#R1">is_constant_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E18:1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rfunct_2.html#T42">RFUNCT_2:42</a></i>;<br/><a NAME="E3:1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="partfun2.html#T57">PARTFUN2:57</a></i>;<br/><a NAME="E4:1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="rolle.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E5:1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rolle.html#E5:1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E8:1_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E9:1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rolle.html#E3:1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="fdiff_1.html#T30">FDIFF_1:30</a></i>;<br/><a NAME="E10:1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rolle.html#E4:1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rolle.html#E8:1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/><b>hence </b><a NAME="E11:1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_2_1"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>and </b><br/><a NAME="E2:1_1_2_1"/><i><font color="Green">E18</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E3:1_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="rolle.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rcomp_1.html#T11">RCOMP_1:11</a></i>;<br/>
<a NAME="E4:1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="rolle.html#E1:1_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rolle.html#E2:1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:1_1_2_1"/><b>then </b>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rolle.html#E3:1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="rolle.html#E1:1_1_2"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_2_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 </span><b>by </b><i><a class="txt" href="rolle.html#E2:1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_1"><b>case </b></a><a NAME="E1:1_1_2_2_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:1_1_2_2_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E4:1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E4:1_1_2_2_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E1:1_1_2_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/><a NAME="E5:1_1_2_2_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rolle.html#E4:1_1_2_2_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:1_1_2_2_1_1"/><i><font color="Green">E21</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:1_1_2_2_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#T10">SEQ_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="funct_1.html#V3">constant</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="rolle.html#E10:1_1_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="seqm_3.html#T15">SEQM_3:15</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E13:1_1_2_2_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_1"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_1_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_1"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E16:1_1_2_2_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><a NAME="E17:1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="seq_4.html#T46">SEQ_4:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="rolle.html#E16:1_1_2_2_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E20:1_1_2_2_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_3"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E2:1_1_2_2_1_1_3"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_2_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_4"/>
 <a href="real_1.html#K1">-</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_1_4"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E23:1_1_2_2_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_1_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/></div><b>end;</b></div><a NAME="E25:1_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="seq_4.html#T46">SEQ_4:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="rolle.html#E23:1_1_2_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E10:1_1_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_1_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_7"/>
0 <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E2:1_1_2_2_1_1_7"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:1_1_2_2_1_1_7"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> 1
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E4:1_1_2_2_1_1_7"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:1_1_2_2_1_1_7"/><i><font color="Green">E32</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E6:1_1_2_2_1_1_7"/>
 <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E7:1_1_2_2_1_1_7"/><b>then </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E5:1_1_2_2_1_1_7"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E8:1_1_2_2_1_1_7"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_2_2_1_1_7"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_1_7"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E10:1_1_2_2_1_1_7"/>
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E29:1_1_2_2_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_2_2_1_1_8"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_2_2_1_1_8"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E3:1_1_2_2_1_1_8"><i><font color="Green">E32</font></i></a>, <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_1"><i><font color="Green">E29</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:1_1_2_2_1_1_8"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E28:1_1_2_2_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_9"/>
0 <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E2:1_1_2_2_1_1_9"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:1_1_2_2_1_1_9"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> 1
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E4:1_1_2_2_1_1_9"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E5:1_1_2_2_1_1_9"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E6:1_1_2_2_1_1_9"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E7:1_1_2_2_1_1_9"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E8:1_1_2_2_1_1_9"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_2_2_1_1_9"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_1_9"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E10:1_1_2_2_1_1_9"/>
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E31:1_1_2_2_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_2_2_1_1_10"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_2_2_1_1_10"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E3:1_1_2_2_1_1_10"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:1_1_2_2_1_1_10"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E30:1_1_2_2_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:1_1_2_2_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rolle.html#E2:1_1_2_2_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rolle.html#E5:1_1_2_2_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fdiff_1.html#T20">FDIFF_1:20</a></i>;<br/><a NAME="E33:1_1_2_2_1_1"/><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rolle.html#E2:1_1_2_2_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rolle.html#E5:1_1_2_2_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rolle.html#E29:1_1_2_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fdiff_1.html#T20">FDIFF_1:20</a></i>;<br/><a NAME="E34:1_1_2_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <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="1_1_2_2_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:1_1_2_2_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E2:1_1_2_2_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E29:1_1_2_2_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:1_1_2_2_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E4:1_1_2_2_1_1_11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rolle.html#E7:1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:1_1_2_2_1_1_11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E19:1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E6:1_1_2_2_1_1_11"/><b>then </b><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E7:1_1_2_2_1_1_11"/><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E14:1_1_2_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E8:1_1_2_2_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/>
<a NAME="E9:1_1_2_2_1_1_11"/><b>then </b><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1_11"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:1_1_2_2_1_1_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E29:1_1_2_2_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_2_1_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:1_1_2_2_1_1_11"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T19">RFUNCT_2:19</a></i>;<br/>
<a NAME="E12:1_1_2_2_1_1_11"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:1_1_2_2_1_1_11"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E12:1_1_2_2_1_1_11"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_1_11"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_1"><i><font color="Green">E29</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E15:1_1_2_2_1_1_11"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E6:1_1_2_2_1_1_11"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rolle.html#E9:1_1_2_2_1_1_11"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:1_1_2_2_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E33:1_1_2_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#T31">SEQ_2:31</a></i>;<br/><a NAME="E36:1_1_2_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:1_1_2_2_1_1_12"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E2:1_1_2_2_1_1_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E31:1_1_2_2_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E3:1_1_2_2_1_1_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E4:1_1_2_2_1_1_12"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rolle.html#E7:1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:1_1_2_2_1_1_12"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E19:1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E6:1_1_2_2_1_1_12"/><b>then </b><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E7:1_1_2_2_1_1_12"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="rolle.html#E21:1_1_2_2_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E8:1_1_2_2_1_1_12"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/>
<a NAME="E9:1_1_2_2_1_1_12"/>
 <a href="real_1.html#K1">-</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_1_12"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E10:1_1_2_2_1_1_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E11:1_1_2_2_1_1_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E12:1_1_2_2_1_1_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T189">XCMPLX_1:189</a></i>;<br/>
<a NAME="E13:1_1_2_2_1_1_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E14:1_1_2_2_1_1_12"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="rolle.html#E8:1_1_2_2_1_1_12"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_1_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:1_1_2_2_1_1_12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E31:1_1_2_2_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_2_1_1_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:1_1_2_2_1_1_12"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T19">RFUNCT_2:19</a></i>;<br/>
<a NAME="E17:1_1_2_2_1_1_12"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E18:1_1_2_2_1_1_12"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E17:1_1_2_2_1_1_12"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E18:1_1_2_2_1_1_12"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_1"><i><font color="Green">E29</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E20:1_1_2_2_1_1_12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E6:1_1_2_2_1_1_12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rolle.html#E14:1_1_2_2_1_1_12"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E37:1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rolle.html#E1:1_1_2_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rolle.html#E32:1_1_2_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rolle.html#E35:1_1_2_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rfunct_2.html#T18">RFUNCT_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_2"><b>case </b></a><a NAME="E1:1_1_2_2_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:1_1_2_2_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E4:1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E4:1_1_2_2_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E1:1_1_2_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rcomp_1.html#T42">RCOMP_1:42</a></i>;<br/><a NAME="E5:1_1_2_2_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rolle.html#E4:1_1_2_2_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:1_1_2_2_1_2"/><i><font color="Green">E21</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:1_1_2_2_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#T10">SEQ_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="funct_1.html#V3">constant</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="rolle.html#E10:1_1_2_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="seqm_3.html#T15">SEQM_3:15</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E13:1_1_2_2_1_2"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_2_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E16:1_1_2_2_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><a NAME="E17:1_1_2_2_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="seq_4.html#T46">SEQ_4:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="rolle.html#E16:1_1_2_2_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E20:1_1_2_2_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_3"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E2:1_1_2_2_1_2_3"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_2_1_2_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_4"/>
 <a href="real_1.html#K1">-</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_2_4"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E23:1_1_2_2_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_2_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/></div><b>end;</b></div><a NAME="E25:1_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="seq_4.html#T46">SEQ_4:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="txt" href="rolle.html#E23:1_1_2_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E10:1_1_2_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><b>hence </b><a NAME="E2:1_1_2_2_1_2_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_7"/>
0 <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E2:1_1_2_2_1_2_7"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:1_1_2_2_1_2_7"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> 1
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E4:1_1_2_2_1_2_7"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:1_1_2_2_1_2_7"/><i><font color="Green">E32</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E6:1_1_2_2_1_2_7"/>
 <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E7:1_1_2_2_1_2_7"/><b>then </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E5:1_1_2_2_1_2_7"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E8:1_1_2_2_1_2_7"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_2_2_1_2_7"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_2_7"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E10:1_1_2_2_1_2_7"/>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E29:1_1_2_2_1_2"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_2_2_1_2_8"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_2_2_1_2_8"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E3:1_1_2_2_1_2_8"><i><font color="Green">E32</font></i></a>, <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_2"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_2"><i><font color="Green">E29</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:1_1_2_2_1_2_8"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E28:1_1_2_2_1_2"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_9"/>
0 <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E2:1_1_2_2_1_2_9"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:1_1_2_2_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> 1
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E4:1_1_2_2_1_2_9"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E5:1_1_2_2_1_2_9"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E6:1_1_2_2_1_2_9"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E7:1_1_2_2_1_2_9"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E8:1_1_2_2_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_2_2_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_2_9"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E10:1_1_2_2_1_2_9"/>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E31:1_1_2_2_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_2_2_1_2_10"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_2_2_1_2_10"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E3:1_1_2_2_1_2_10"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E20:1_1_2_2_1_2"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_2"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> 2</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:1_1_2_2_1_2_10"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E30:1_1_2_2_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:1_1_2_2_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rolle.html#E2:1_1_2_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rolle.html#E5:1_1_2_2_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fdiff_1.html#T20">FDIFF_1:20</a></i>;<br/><a NAME="E33:1_1_2_2_1_2"/><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rolle.html#E2:1_1_2_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rolle.html#E5:1_1_2_2_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rolle.html#E29:1_1_2_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fdiff_1.html#T20">FDIFF_1:20</a></i>;<br/><a NAME="E34:1_1_2_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:1_1_2_2_1_2_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E2:1_1_2_2_1_2_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E29:1_1_2_2_1_2"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:1_1_2_2_1_2_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E4:1_1_2_2_1_2_11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rolle.html#E7:1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:1_1_2_2_1_2_11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E19:1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E6:1_1_2_2_1_2_11"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E7:1_1_2_2_1_2_11"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E8:1_1_2_2_1_2_11"/><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E14:1_1_2_2_1_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E9:1_1_2_2_1_2_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/>
<a NAME="E10:1_1_2_2_1_2_11"/><b>then </b><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E8:1_1_2_2_1_2_11"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:1_1_2_2_1_2_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E29:1_1_2_2_1_2"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_2_1_2_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:1_1_2_2_1_2_11"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T19">RFUNCT_2:19</a></i>;<br/>
<a NAME="E13:1_1_2_2_1_2_11"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:1_1_2_2_1_2_11"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E13:1_1_2_2_1_2_11"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E14:1_1_2_2_1_2_11"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_2"><i><font color="Green">E29</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E16:1_1_2_2_1_2_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2_11"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_2_11"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:1_1_2_2_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="rolle.html#E33:1_1_2_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rfunct_2.html#T18">RFUNCT_2:18</a></i>;<br/><a NAME="E36:1_1_2_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <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="1_1_2_2_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:1_1_2_2_1_2_12"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E2:1_1_2_2_1_2_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E31:1_1_2_2_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E3:1_1_2_2_1_2_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E4:1_1_2_2_1_2_12"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rolle.html#E7:1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:1_1_2_2_1_2_12"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E19:1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E6:1_1_2_2_1_2_12"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E7:1_1_2_2_1_2_12"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E8:1_1_2_2_1_2_12"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="rolle.html#E21:1_1_2_2_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E9:1_1_2_2_1_2_12"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/>
<a NAME="E10:1_1_2_2_1_2_12"/>
 <a href="real_1.html#K1">-</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rolle.html#E8:1_1_2_2_1_2_12"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E11:1_1_2_2_1_2_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E12:1_1_2_2_1_2_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E13:1_1_2_2_1_2_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T189">XCMPLX_1:189</a></i>;<br/>
<a NAME="E14:1_1_2_2_1_2_12"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E15:1_1_2_2_1_2_12"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="rolle.html#E9:1_1_2_2_1_2_12"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_2_12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:1_1_2_2_1_2_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:1_1_2_2_1_2_12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E31:1_1_2_2_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_2_1_2_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="rolle.html#E4:1_1_2_2_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:1_1_2_2_1_2_12"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T19">RFUNCT_2:19</a></i>;<br/>
<a NAME="E18:1_1_2_2_1_2_12"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E8:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:1_1_2_2_1_2_12"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E6:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E12:1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E10:1_1_2_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E18:1_1_2_2_1_2_12"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E19:1_1_2_2_1_2_12"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E27:1_1_2_2_1_2"><i><font color="Green">E29</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E21:1_1_2_2_1_2_12"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="rolle.html#E7:1_1_2_2_1_2_12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rolle.html#E15:1_1_2_2_1_2_12"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T130">XREAL_1:130</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E37:1_1_2_2_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rolle.html#E1:1_1_2_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rolle.html#E32:1_1_2_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rolle.html#E35:1_1_2_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="seq_2.html#T31">SEQ_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
