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<span class="kw">let </span><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> ; <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">i</font> being    <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/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b3">f</font> being   <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="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/>  for <font color="Olive" title="b4">x</font> being   <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>  st <font color="Olive" title="b3">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">i</font> holds <br/>( <font color="Olive" title="b2">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b3">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">i</font> &amp;  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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> ; <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><br/>  for <font color="Olive" title="b2">f</font> being   <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="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/>  for <font color="Olive" title="b3">x</font> being   <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>  st <font color="Olive" title="b2">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c5">i</font> holds <br/>( <font color="Olive" title="b1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b2">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b2">f</font></span>)</span>,<font color="Olive" title="b3">x</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">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">  for <font color="Olive" title="b1">f</font> being   <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="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/>  for <font color="Olive" title="b2">x</font> being   <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>  st <font color="Olive" title="b1">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c5">i</font> holds <br/>( <font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b1">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">f</font> be   <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="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>; <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">x</font> being   <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>  st <font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">i</font> holds <br/>( <font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font></span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.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="c7">f</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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>; <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="c7">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font> implies ( <font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font></span>)</span>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.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="c7">f</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:56"/>
<font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</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="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font></span>)</span>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.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="c7">f</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )</span><br/>

<a NAME="E2:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c7">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="c5">i</font>,<font color="Maroon" title="c8">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="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D9" target="_self" title="PDIFF_1:def.9">Def9</a></span>;<br/>
<a NAME="E3:56"/>
<font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</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="c5">i</font>,<font color="Maroon" title="c8">x</font>)</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</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="c8">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T27" target="_self" title="PDIFF_1:th.27">Th27</a></span>;<br/>
<a NAME="E4:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</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="c8">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="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="ndiff_1.html#T37" title="NDIFF_1:th.37">NDIFF_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56"/>
<font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D9" target="_self" title="PDIFF_1:def.9">Def9</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#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font></span>)</span>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.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="c7">f</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><br/>

<a NAME="E6:56"/>
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</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="c5">i</font>,<font color="Maroon" title="c8">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="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="c8">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="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font></span>)</span>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T27" target="_self" title="PDIFF_1:th.27">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:56"/>
 <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c7">f</font></span>)</span>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.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="c7">f</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="ndiff_1.html#T37" title="NDIFF_1:th.37">NDIFF_1:37</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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