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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>   st <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> holds <br/> <a href="pdiff_3.html#K6" title="PDIFF_3:func.6">hpartdiff22</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_2.html#K4" title="PDIFF_2:func.4">partdiff2</a> <span class="p1">(<span class="default"><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></span>)</span>,<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> implies  <a href="pdiff_3.html#K6" title="PDIFF_3:func.6">hpartdiff22</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_2.html#K4" title="PDIFF_2:func.4">partdiff2</a> <span class="p1">(<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><br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E25">A1</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#K6" title="PDIFF_3:func.6">hpartdiff22</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_2.html#K4" title="PDIFF_2:func.4">partdiff2</a> <span class="p1">(<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><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="E3:32"/><i><font color="Green" title="E26">A2</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>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120" title="FINSEQ_2:th.120">FINSEQ_2:120</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E27">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>
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="pdiff_3.html#T12" target="_self" title="PDIFF_3:th.12">Th12</a></i>;<br/>
 <a href="pdiff_3.html#K6" title="PDIFF_3:func.6">hpartdiff22</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</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#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>,<font color="Maroon" title="c4">y0</font>
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E3:32"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="pdiff_3.html#T8" target="_self" title="PDIFF_3:th.8">Th8</a></i>
<br/>.= 
 <a href="pdiff_2.html#K4" title="PDIFF_2:func.4">partdiff2</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E4:32"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="pdiff_2.html#T12" title="PDIFF_2:th.12">PDIFF_2:12</a></i>
;<br/>
<b>hence </b><a NAME="E6:32"/>
 <a href="pdiff_3.html#K6" title="PDIFF_3:func.6">hpartdiff22</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_2.html#K4" title="PDIFF_2:func.4">partdiff2</a> <span class="p1">(<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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