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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z0</font> being    <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> 2<br/>  for <font color="Olive" title="b2">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> 2</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_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>,1)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</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>,1)</span>)</span></span>)</span>,<font color="Olive" title="b1">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">z0</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">z0</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> 2; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> 2</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_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>,1)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</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>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">z0</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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> 2</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_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>,1)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</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>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font>)</span>)</span> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c2">z0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>,1)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</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>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font>)</span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c1">r</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E3:38"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c3">f</font>,1) <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z0</font>,2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_3.html#T10" target="_self" onmouseover="rs('pdiff_3/T10')" onmouseout="rh()">Th10</a>, <a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/>
<a NAME="E4:38"/>
 <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">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>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</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>,1)</span>)</span>,<font color="Maroon" title="c2">z0</font>,2)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T33" onmouseover="rs('pdiff_1/T33')" onmouseout="rh()">PDIFF_1:33</a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:38"/>
( <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>,1)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</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>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c2">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">z0</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="pdiff_3.html#T14" target="_self" onmouseover="rs('pdiff_3/T14')" onmouseout="rh()">Th14</a>, <a class="ref" href="pdiff_1.html#T33" onmouseover="rs('pdiff_1/T33')" onmouseout="rh()">PDIFF_1:33</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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