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<b>let </b><font color="Maroon" title="c1">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="c2">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="c3">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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">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="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">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="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">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="c1">n</font>;<br/>











<b>assume </b><a NAME="E1:27"/>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c3">f</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="c2">i</font>,<font color="Maroon" title="c5">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"><font color="Maroon" title="c4">g</font> <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="c2">i</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> &amp; <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="c2">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T13" target="_self" title="PDIFF_1:th.13">Th13</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="E1:27_1"/>
<font color="Maroon" title="c3">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">i</font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
<font color="Maroon" title="c3">f</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="c2">i</font>,<font color="Maroon" title="c5">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="c2">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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="E3:27_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <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="c2">i</font>,<font color="Maroon" title="c6">y</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="c2">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E2:27"><i><font color="Green" title="E22">A1</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="E4:27_1"/>
<font color="Maroon" title="c4">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="ref" href="pdiff_1.html#D11" target="_self" title="PDIFF_1:def.11">Def11</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:27"/>
<font color="Maroon" title="c4">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">i</font>
 ;<br/>

<a NAME="E5:27"/><b>then </b>
<font color="Maroon" title="c4">g</font> <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="c2">i</font>,<font color="Maroon" title="c6">y</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="c2">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#D11" target="_self" title="PDIFF_1:def.11">Def11</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
<font color="Maroon" title="c3">f</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="c2">i</font>,<font color="Maroon" title="c5">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="c2">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E2:27"><i><font color="Green" title="E22">A1</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="E7:27"/>
<font color="Maroon" title="c3">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">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/>


</div>
