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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">y0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">m</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b2">y0</font> holds <br/>( <font color="Olive" title="b3">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b1">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b2">y0</font> &amp;  <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b1">g</font></span>)</span>,<font color="Olive" title="b2">y0</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">y0</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">y0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">m</font><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">y0</font> holds <br/>( <font color="Olive" title="b2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">y0</font> &amp;  <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Olive" title="b1">y0</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c3">g</font>,<font color="Olive" title="b1">y0</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">y0</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">m</font>; <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">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">y0</font> holds <br/>( <font color="Olive" title="b1">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">y0</font> &amp;  <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">y0</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">y0</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">y0</font> implies ( <font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">y0</font> &amp;  <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">y0</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">y0</font>)</span>)</span> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">y0</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="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">y0</font> &amp;  <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">y0</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">y0</font>)</span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c3">g</font> as   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#T1" target="_self" title="PDIFF_6:th.1">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x0</font> = <font color="Maroon" title="c4">y0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E4:30"/>
<font color="Maroon" title="c6">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="pdiff_6.html#T2" target="_self" title="PDIFF_6:th.2">Th2</a></span>;<br/>
<a NAME="E5:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A2</font></span>: 
( <font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c6">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">x0</font> &amp;  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c6">f</font></span>)</span>,<font color="Maroon" title="c7">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">x0</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#T37" title="NDIFF_1:th.37">NDIFF_1:37</a></span>;<br/>
<a NAME="E6:30"/>
<font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#T6" target="_self" title="PDIFF_6:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:30"/>
<font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">y0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="pdiff_6.html#T2" target="_self" title="PDIFF_6:th.2">Th2</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="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">y0</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">y0</font>)</span>)</span></span><br/>

<a NAME="E8:30"/><span class="kw">then </span>
 <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">y0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c6">f</font></span>)</span>,<font color="Maroon" title="c7">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#T3" target="_self" title="PDIFF_6:th.3">Th3</a>, <a class="ref" href="pdiff_6.html#T6" target="_self" title="PDIFF_6:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:30"/>
 <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">y0</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">y0</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="pdiff_6.html#T9" target="_self" title="PDIFF_6:th.9">Th9</a>, <a class="ref" href="pdiff_6.html#T3" target="_self" title="PDIFF_6:th.3">Th3</a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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