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<span class="kw">let </span><font color="Maroon" title="c3">x0</font>, <font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</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">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font> holds <br/>( <font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c3">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">x0</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font> implies ( <font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">x0</font>)</span>)</span> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">x0</font>)</span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">N1</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c3">x0</font><span class="kw"> such that </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c6">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 ex <font color="Olive" title="b1">L</font> being   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b2">R</font> being   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/> for <font color="Olive" title="b3">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D4" target="_self" title="FDIFF_1:def.4">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">L1</font> being   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a>, <font color="Maroon" title="c8">R1</font> being   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a><span class="kw"> such that </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">R</font> = <font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c8">R1</font> as   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#T5" target="_self" title="FDIFF_1:th.5">Th9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">L</font> = <font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c7">L1</font> as   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#T3" target="_self" title="FDIFF_1:th.3">Th7</a></span>;<br/>
<a NAME="E9:28"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon" title="c7">L1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D3" target="_self" title="FDIFF_1:def.3">Def4</a></span>;<br/>
<a NAME="E10:28"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c6">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E11:28"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c8">R1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D2" target="_self" title="FDIFF_1:def.2">Def3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E28">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> )</span><br/><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></span>;<br/><span class="kw">assume </span><a NAME="E2:28_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="rfunct_1.html#T57" title="RFUNCT_1:th.57">RFUNCT_1:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="rfunct_1.html#T57" title="RFUNCT_1:th.57">RFUNCT_1:57</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:28"/>
<font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="fdiff_1.html#D4" target="_self" title="FDIFF_1:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">x0</font>)</span>)</span></span><br/>

<span class="kw">hence </span> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c3">x0</font>) = 
<font color="Maroon" title="c10">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="fdiff_1.html#D5" target="_self" title="FDIFF_1:def.5">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="rfunct_1.html#T57" title="RFUNCT_1:th.57">RFUNCT_1:57</a></span>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">x0</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="fdiff_1.html#D5" target="_self" title="FDIFF_1:def.5">Def6</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
