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

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">Z</font> holds <br/>for <font color="Olive" title="b3">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Olive" title="b3">m</font>,<font color="Olive" title="b1">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">Z</font> holds <br/>for <font color="Olive" title="b2">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c2">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">Z</font> implies for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">Z</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:34"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">Z</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">m</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> implies <font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">Z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:34"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">Z</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:34_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 implies <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font> )</span><br/><span class="kw">assume </span><a NAME="E2:34_1"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font></span><br/><a NAME="E3:34_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:34_1"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="taylor_1.html#D6" target="_self" title="TAYLOR_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:34"/>
<font color="Maroon" title="c1">f</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_1.html#D6" target="_self" title="TAYLOR_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
