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<b>let </b><font color="Maroon" title="c1">f1</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"> 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>   st <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> &amp; <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> holds <br/>(  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Olive" title="b2">x</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> &amp; <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font> implies (  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font> &amp; ( 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="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Olive" title="b1">x</font></span>)</span> ) ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font> &amp; ( 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="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Olive" title="b1">x</font></span>)</span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> implies  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font> )</span><br/><b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font></span><br/><a NAME="E2:27_1"/><b>then </b>
<font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="txt" href="integra8.html#E2:27"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="ref" href="integra8.html#T22" target="_self" title="INTEGRA8:th.22">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:27"/><i><font color="Green" title="E30">A3</font></i>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="integra8.html#E1:27"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>; <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">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="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Olive" title="b1">x</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:27_2"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</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"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x</font></span>)</span></span><br/><a NAME="E2:27_2"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="integra8.html#E2:27"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span>,<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="integra8.html#E4:27"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="integra8.html#E1:27_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="integra8.html#E2:27_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="integra8.html#T22" target="_self" title="INTEGRA8:th.22">Th22</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:27"/>
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="c2">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Olive" title="b1">x</font></span>)</span>
 ; <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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