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<b>let </b><font color="Maroon">c<sub>1</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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <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/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 1,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E2:49_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:49_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:49_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:49_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:49"><i><font color="Green">E44</font></i></a>, <a class="ref" href="taylor_1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E5:49_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:49"><i><font color="Green">E44</font></i></a>, <a class="txt" href="taylor_1.html#E1:49_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:49_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/><a NAME="E7:49_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>;<br/><b>hence </b><a NAME="E8:49_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">diff</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:49"><i><font color="Green">E43</font></i></a>, <a class="txt" href="taylor_1.html#E4:49_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="taylor_1.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
<font color="Maroon">c<sub>1</sub></font> <a href="taylor_1.html#R1" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 1,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="taylor_1.html#D5" target="_self">Def5</a></i>;<br/>


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