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<b>let </b><font color="Maroon" title="c1">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z</font> iff  <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font> )</span><br/><b>let </b><font color="Maroon" title="c2">f</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> 2, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z</font> iff  <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font> )</span><br/>



<b>thus </b><a NAME="E1:24"/>
( <font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z</font> implies  <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font> implies <font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green" title="E21">A0</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font></span><br/>

<b>consider </b><font color="Maroon" title="c3">x0</font>, <font color="Maroon" title="c4">y0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b1">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">y0</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font></span>)</span>,<font color="Maroon" title="c1">z</font></span>)</span> &amp;  ex <font color="Olive" title="b2">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b3">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>for <font color="Olive" title="b4">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font></span>)</span>,<font color="Maroon" title="c1">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font></span>)</span>,<font color="Maroon" title="c1">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">y0</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A0</font></i></a>, <a class="ref" href="pdiff_3.html#D6" target="_self" title="PDIFF_3:def.6">Def6</a></i>;<br/>
<b>thus </b><a NAME="E4:24_1"/>
 <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font>
 <b>by </b><i><a class="txt" href="#E3:24_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="pdiff_2.html#D6" title="PDIFF_2:def.6">PDIFF_2:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:24"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z</font></span><br/>

<b>consider </b><font color="Maroon" title="c3">x0</font>, <font color="Maroon" title="c4">y0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:24"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b1">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">y0</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font></span>)</span>,<font color="Maroon" title="c1">z</font></span>)</span> &amp;  ex <font color="Olive" title="b2">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b3">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>for <font color="Olive" title="b4">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font></span>)</span>,<font color="Maroon" title="c1">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font></span>)</span>,<font color="Maroon" title="c1">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">y0</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="pdiff_2.html#D6" title="PDIFF_2:def.6">PDIFF_2:def 6</a></i>;<br/>
<b>thus </b><a NAME="E5:24"/>
<font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z</font>
 <b>by </b><i><a class="txt" href="#E4:24"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="pdiff_3.html#D6" target="_self" title="PDIFF_3:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
