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<span class="kw">let </span><font color="Maroon" title="c1">u</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#R4" title="PDIFF_5:pred.4">is_hpartial_differentiable`21_in</a> <font color="Maroon" title="c1">u</font> holds <br/> <a href="pdiff_5.html#K4" title="PDIFF_5:func.4">hpartdiff21</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b1">f</font>,2</span>)</span>,<font color="Maroon" title="c1">u</font>,1</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#R4" title="PDIFF_5:pred.4">is_hpartial_differentiable`21_in</a> <font color="Maroon" title="c1">u</font> implies  <a href="pdiff_5.html#K4" title="PDIFF_5:func.4">hpartdiff21</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,2</span>)</span>,<font color="Maroon" title="c1">u</font>,1 )</span><br/>



<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="pdiff_5.html#R4" title="PDIFF_5:pred.4">is_hpartial_differentiable`21_in</a> <font color="Maroon" title="c1">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pdiff_5.html#K4" title="PDIFF_5:func.4">hpartdiff21</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,2</span>)</span>,<font color="Maroon" title="c1">u</font>,1</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">x0</font>, <font color="Maroon" title="c4">y0</font>, <font color="Maroon" title="c5">z0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">y0</font>,<font color="Maroon" title="c5">z0</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T123" title="FINSEQ_2:th.123">FINSEQ_2:123</a></span>;<br/>
 <a href="pdiff_5.html#K4" title="PDIFF_5:func.4">hpartdiff21</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">u</font> = 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 1,<span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,2</span>)</span>,<font color="Maroon" title="c1">u</font></span>)</span>,<font color="Maroon" title="c3">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="pdiff_5.html#T13" target="_self" title="PDIFF_5:th.13">Th7</a></span>
<br/>.= 
 <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,2</span>)</span>,<font color="Maroon" title="c1">u</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="pdiff_4.html#T19" title="PDIFF_4:th.19">PDIFF_4:19</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:51"/>
 <a href="pdiff_5.html#K4" title="PDIFF_5:func.4">hpartdiff21</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,2</span>)</span>,<font color="Maroon" title="c1">u</font>,1
 ; <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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