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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span><br/>  for <font color="Olive" title="b2">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b3">f</font> <a href="pdiff_9.html#R4" title="PDIFF_9:pred.4">is_partial_differentiable_on</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">I</font> holds <br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="pdiff_9.html#K5" title="PDIFF_9:func.5">`partial|</a> (<font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">I</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">f</font> <a href="pdiff_9.html#R4" title="PDIFF_9:pred.4">is_partial_differentiable_on</a> <font color="Maroon" title="c2">Z</font>,<font color="Olive" title="b1">I</font> holds <br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="pdiff_9.html#K5" title="PDIFF_9:func.5">`partial|</a> (<font color="Maroon" title="c2">Z</font>,<font color="Olive" title="b1">I</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">I</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">f</font> <a href="pdiff_9.html#R4" title="PDIFF_9:pred.4">is_partial_differentiable_on</a> <font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font> holds <br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="pdiff_9.html#K5" title="PDIFF_9:func.5">`partial|</a> (<font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="pdiff_9.html#R4" title="PDIFF_9:pred.4">is_partial_differentiable_on</a> <font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font> implies  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pdiff_9.html#K5" title="PDIFF_9:func.5">`partial|</a> (<font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font> )</span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a>, <a class="ref" href="finseq_1.html#T20" onmouseover="rs('finseq_1/T20')" onmouseout="rh()">FINSEQ_1:20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:2"/>
<font color="Maroon" title="c4">f</font> <a href="pdiff_9.html#R4" title="PDIFF_9:pred.4">is_partial_differentiable_on</a> <font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pdiff_9.html#K5" title="PDIFF_9:func.5">`partial|</a> (<font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font></span><br/>

<a NAME="E3:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<span class="p1">(<span class="default"><a href="pdiff_9.html#K4" title="PDIFF_9:func.4">PartDiffSeq</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="pdiff_9.html#R3" title="PDIFF_9:pred.3">is_partial_differentiable_on</a> <font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E4:2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_9.html#K4" title="PDIFF_9:func.4">PartDiffSeq</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_9.html#K4" title="PDIFF_9:func.4">PartDiffSeq</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="pdiff_9.html#K3" title="PDIFF_9:func.3">`partial|</a> (<font color="Maroon" title="c2">Z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_9.html#D7" onmouseover="rs('pdiff_9/D7')" onmouseout="rh()">PDIFF_9:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pdiff_9.html#K5" title="PDIFF_9:func.5">`partial|</a> (<font color="Maroon" title="c2">Z</font>,<font color="Maroon" title="c3">I</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="pdiff_9.html#D6" onmouseover="rs('pdiff_9/D6')" onmouseout="rh()">PDIFF_9:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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