<?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="E1:8"/><i><font color="Green">E5</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>, <font color="Maroon">c<sub>4</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="E2:8"/><i><font color="Green">E6</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>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</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>
 <b>and </b><br/><a NAME="E4:8"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#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> holds <br/> <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>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</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="E6:8"/><i><font color="Green">E9</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#E2:8"><i><font color="Green">E6</font></i></a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>
<a NAME="E7:8"/>
<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>4</sub></font>
 
<b>by </b><i><a class="txt" href="rolle.html#E3:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>
<a NAME="E8:8"/><b>then </b>
<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">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rolle.html#E6:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E9:8"/><b>then </b><i><font color="Green">E10</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E10:8"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><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#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="rolle.html#E2:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E3:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</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/><b>hence </b> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> = 
<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#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="rolle.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E2:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rolle.html#E3:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rolle.html#E9:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E1:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E4:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rolle.html#E1:8_1"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
0

;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:8"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun2.html#R1">is_constant_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>
 <b>by </b><i><a class="txt" href="rolle.html#E1:8"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rolle.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rolle.html#T7" target="_self">Th7</a></i>;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:8"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="partfun2.html#T76">PARTFUN2:76</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E15:8"/><i><font color="Green">E13</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/>

<a NAME="E16:8"/><b>then </b>
<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rolle.html#E9:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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#E14:8"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<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>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rolle.html#E9:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rolle.html#E15:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E18:8"/>
<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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div>
