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<div class="add">

<b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">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="c5">i</font>, <font color="Maroon" title="c6">j</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="c7">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="c4">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="c3">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">h</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="c4">m</font>, <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">n</font>;<br/><b>let </b><font color="Maroon" title="c9">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="c4">m</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c10">z</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="c4">m</font>;<br/>















<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> )
 ;<br/>

<a NAME="E2:42"/><b>then </b><i><font color="Green" title="E32">A2</font></i>: 
<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="c6">j</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f</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="c5">i</font>,<font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K2" title="PDIFF_1:func.2">&lt;&gt;*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c10">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T22" target="_self" title="PDIFF_1:th.22">Th22</a></i>;<br/>
<a NAME="E3:42"/><i><font color="Green" title="E33">A3</font></i>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">z</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:42"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="pdiff_1.html#D4" target="_self" title="PDIFF_1:def.4">Def4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 ;<br/><a NAME="E2:42_1"/><b>then </b>
<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="c6">j</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f</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="c5">i</font>,<font color="Maroon" title="c9">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="c5">i</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="pdiff_1.html#D15" target="_self" title="PDIFF_1:def.15">Def15</a></i>;<br/><a NAME="E3:42_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c10">z</font></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E2:42"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="pdiff_1.html#E3:42"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="pdiff_1.html#T7" target="_self" title="PDIFF_1:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E4:42_1"/>
<font color="Maroon" title="c8">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="ref" href="pdiff_1.html#D17" target="_self" title="PDIFF_1:def.17">Def17</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:42"/>
<font color="Maroon" title="c8">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 ;<br/>

<a NAME="E6:42"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c10">z</font></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#D17" target="_self" title="PDIFF_1:def.17">Def17</a></i>;<br/>
<a NAME="E7:42"/><b>then </b>
<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="c6">j</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f</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="c5">i</font>,<font color="Maroon" title="c9">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="c5">i</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E2:42"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="pdiff_1.html#E3:42"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="pdiff_1.html#T8" target="_self" title="PDIFF_1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E8:42"/>
<font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="ref" href="pdiff_1.html#D15" target="_self" title="PDIFF_1:def.15">Def15</a></i>;<br/>


</div>
