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<b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">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="c3">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="c4">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">G</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">m</font>, <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font>;<br/><b>let </b><font color="Maroon" title="c7">x</font> be   <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="c3">m</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">y</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="c3">m</font>;<br/>









<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c5">F</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">x</font> )
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T20" target="_self" title="PDIFF_1:th.20">Th20</a></i>;<br/>
<a NAME="E3:40"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">m</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></i>;<br/>
<a NAME="E4:40"/><b>then </b>
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c7">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">m</font>, <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:40"/>
 <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="pdiff_1.html#E2:40"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="pdiff_1.html#D8" target="_self" title="PDIFF_1:def.8">Def8</a></i>;<br/>


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