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<span class="kw">let </span><font color="Maroon" title="c1">u0</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> 3; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> 3</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font> <a href="pdiff_5.html#R6" title="PDIFF_5:pred.6">is_hpartial_differentiable`23_in</a> <font color="Maroon" title="c1">u0</font> holds <br/> ex <font color="Olive" title="b2">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Olive" title="b2">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">R</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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> 3</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="pdiff_5.html#R6" title="PDIFF_5:pred.6">is_hpartial_differentiable`23_in</a> <font color="Maroon" title="c1">u0</font> implies  ex <font color="Olive" title="b1">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Olive" title="b1">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">R</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:110"/>
<font color="Maroon" title="c2">f</font> <a href="pdiff_5.html#R6" title="PDIFF_5:pred.6">is_hpartial_differentiable`23_in</a> <font color="Maroon" title="c1">u0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Olive" title="b1">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">R</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E2:110"/><span class="kw">then </span>
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,2 <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_5.html#T24" target="_self" title="PDIFF_5:th.24">Th12ForZ</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:110"/>
 ex <font color="Olive" title="b1">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Olive" title="b1">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">R</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_4.html#T36" title="PDIFF_4:th.36">PDIFF_4:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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