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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <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">f1</font>, <font color="Olive" title="b3">f2</font> being  <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>   st <font color="Olive" title="b2">f1</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b3">f2</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">f2</font></span>)</span>,<font color="Olive" title="b1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Olive" title="b3">f2</font>,<font color="Olive" title="b1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <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>   st <font color="Olive" title="b1">f1</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Z</font> &amp; <font color="Olive" title="b2">f2</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Z</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">f2</font></span>)</span>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f1</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Z</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Z</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Z</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c5">f2</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> or <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E3:18"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span></span><br/>

<a NAME="E4:18"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">Z</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T23" title="TAYLOR_1:th.23">TAYLOR_1:23</a>, <a class="txt" href="hfdiff_1.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="hfdiff_1.html#E3:18"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<a NAME="E5:18"/>
<font color="Maroon" title="c5">f2</font> <a href="taylor_1.html#R1" title="TAYLOR_1:pred.1">is_differentiable_on</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">Z</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T23" title="TAYLOR_1:th.23">TAYLOR_1:23</a>, <a class="txt" href="hfdiff_1.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="hfdiff_1.html#E3:18"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:18"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="hfdiff_1.html#T15" target="_self" title="HFDIFF_1:th.15">Th7</a>, <a class="txt" href="hfdiff_1.html#E4:18"><i><font color="Green" title="E17">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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