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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <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">z0</font> being   <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<br/> for <font color="Olive" title="b2">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="b2">f</font> <a href="pdiff_3.html#R1" title="PDIFF_3:pred.1">is_hpartial_differentiable`11_in</a> <font color="Olive" title="b1">z0</font> holds <br/>( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">z0</font></span>)</span> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Olive" title="b1">z0</font> &amp;  <a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">z0</font></span>)</span></span>)</span>,<font color="Olive" title="b1">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff11</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">z0</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">z0</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#R1" title="PDIFF_3:pred.1">is_hpartial_differentiable`11_in</a> <font color="Maroon" title="c2">z0</font> holds <br/>( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</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>,<font color="Maroon" title="c2">z0</font></span>)</span> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z0</font> &amp;  <a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff11</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">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="c3">f</font> <a href="pdiff_3.html#R1" title="PDIFF_3:pred.1">is_hpartial_differentiable`11_in</a> <font color="Maroon" title="c2">z0</font> implies ( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z0</font> &amp;  <a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff11</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> ) )</span><br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="pdiff_3.html#R1" title="PDIFF_3:pred.1">is_hpartial_differentiable`11_in</a> <font color="Maroon" title="c2">z0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z0</font> &amp;  <a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff11</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> )</span><br/>

<a NAME="E2:45"/><b>then </b>
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z0</font>
 
<b>by </b><i><a class="ref" href="pdiff_3.html#T9" target="_self" title="PDIFF_3:th.9">Th9</a></i>;<br/>
<a NAME="E3:45"/><b>then </b>
( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z0</font> &amp;  <a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pdiff_2.html#T21" title="PDIFF_2:th.21">PDIFF_2:21</a></i>;<br/>
<b>hence </b><a NAME="E4:45"/>
( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z0</font> &amp;  <a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff11</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="pdiff_3.html#T17" target="_self" title="PDIFF_3:th.17">Th17</a></i>; <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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