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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font></span>)</span><br/>  for <font color="Olive" title="b3">Z</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>  st <font color="Olive" title="b3">Z</font> is  <a href="pdiff_7.html#V1" title="PDIFF_7:attr.1">open</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Olive" title="b2">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Olive" title="b3">Z</font>,<font color="Olive" title="b1">i</font> iff ( <font color="Olive" title="b3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> &amp; (  for <font color="Olive" title="b4">x</font> being    <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">m</font>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Z</font> holds <br/><font color="Olive" title="b2">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">i</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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">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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font></span>)</span><br/>  for <font color="Olive" title="b2">Z</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>  st <font color="Olive" title="b2">Z</font> is  <a href="pdiff_7.html#V1" title="PDIFF_7:attr.1">open</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Olive" title="b2">Z</font>,<font color="Maroon" title="c3">i</font> iff ( <font color="Olive" title="b2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; (  for <font color="Olive" title="b3">x</font> being    <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">m</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Z</font> holds <br/><font color="Olive" title="b1">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c3">i</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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">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">Z</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>  st <font color="Olive" title="b1">Z</font> is  <a href="pdiff_7.html#V1" title="PDIFF_7:attr.1">open</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Maroon" title="c4">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Olive" title="b1">Z</font>,<font color="Maroon" title="c3">i</font> iff ( <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <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">m</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c3">i</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">Z</font> is  <a href="pdiff_7.html#V1" title="PDIFF_7:attr.1">open</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> implies ( <font color="Maroon" title="c4">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c3">i</font> iff ( <font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <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">m</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">i</font> ) ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
( <font color="Maroon" title="c5">Z</font> is  <a href="pdiff_7.html#V1" title="PDIFF_7:attr.1">open</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</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="c4">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c3">i</font> iff ( <font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <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">m</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">i</font> ) ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Z0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">m</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
( <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z0</font> &amp; <font color="Maroon" title="c6">Z0</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#D3" target="_self" title="PDIFF_7:def.3">Def3</a></span>;<br/>
<a NAME="E4:51"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c4">f</font> as   <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="c1">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="c2">n</font></span>)</span> ;<br/>
<div><span class="kw">hereby </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="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <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">m</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">i</font> ) implies <font color="Maroon" title="c4">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c3">i</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:51_1"/>
<font color="Maroon" title="c4">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c3">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="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <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">m</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">i</font> ) )</span><br/><a NAME="E2:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Maroon" title="c6">Z0</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="pdiff_7.html#T33" target="_self" title="PDIFF_7:th.33">Th33</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1"/>
<font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="pdiff_7.html#T8" target="_self" title="PDIFF_7:th.8">Th8</a></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="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">m</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">i</font></span><br/><span class="kw">thus </span><a NAME="E4:51_1"/>
 for <font color="Olive" title="b1">x</font> being    <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">m</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">i</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</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">m</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> implies <font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</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="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">i</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">x</font> as   <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">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E3:51_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="pdiff_7.html#T8" target="_self" title="PDIFF_7:th.8">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:51_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D13" title="PDIFF_1:def.13">PDIFF_1:def 13</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/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E7:51"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
( <font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <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">m</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">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="c4">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c3">i</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">y</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">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="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z0</font> implies <font color="Maroon" title="c7">g</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c3">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:51_2"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z0</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="c7">g</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c3">i</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">y</font> as    <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">m</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/><a NAME="E3:51_2"/>
<font color="Maroon" title="c4">f</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E48">A5</font></span></a></span>;<br/><a NAME="E4:51_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">gZ</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="c1">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="c2">n</font></span>)</span> ex <font color="Olive" title="b2">y</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="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">gZ</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">gZ</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c3">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D13" title="PDIFF_1:def.13">PDIFF_1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_2"/>
<font color="Maroon" title="c7">g</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:51"/><span class="kw">then </span>
<font color="Maroon" title="c7">g</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Maroon" title="c6">Z0</font>,<font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="pdiff_7.html#T8" target="_self" title="PDIFF_7:th.8">Th8</a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E48">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:51"/>
<font color="Maroon" title="c4">f</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#T33" target="_self" title="PDIFF_7:th.33">Th33</a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E47">A2</font></span></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/>


</div>
