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<b>let </b><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#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="c3">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> ;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</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="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>;<br/><b>let </b><font color="Maroon" title="c6">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="c2">m</font></span>)</span>;<br/>











<b>assume </b><a NAME="E1:51"/>
( <font color="Maroon" title="c4">f1</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">i</font> )
 ;<br/>

<a NAME="E2:51"/><b>then </b><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="ref" href="pdiff_1.html#D9" target="_self" title="PDIFF_1:def.9">Def9</a></i>;<br/>
<a NAME="E3:51"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T26" target="_self" title="PDIFF_1:th.26">Th26</a></i>;<br/>
<a NAME="E4:51"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E2:51"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="ndiff_1.html#T41" title="NDIFF_1:th.41">NDIFF_1:41</a></i>;<br/>
<b>hence </b><a NAME="E5:51"/>
<font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="ref" href="pdiff_1.html#D9" target="_self" title="PDIFF_1:def.9">Def9</a></i>;<br/>

<a NAME="E6:51"/>
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T26" target="_self" title="PDIFF_1:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E7:51"/>
 <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E2:51"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="ndiff_1.html#T41" title="NDIFF_1:th.41">NDIFF_1:41</a></i>;<br/>


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