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<span class="kw">let </span><font color="Maroon" title="c1">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">f1</font>, <font color="Olive" title="b2">f2</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">f1</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z0</font> &amp; <font color="Olive" title="b2">f2</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z0</font> holds <br/>( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Olive" title="b1">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Olive" title="b1">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span></span>)</span>,<font color="Maroon" title="c1">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</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="c2">f1</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z0</font> implies ( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span></span>)</span>,<font color="Maroon" title="c1">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">z0</font>
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c3">f2</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c1">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"> ( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span></span>)</span>,<font color="Maroon" title="c1">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> )</span><br/>

<a NAME="E3:32"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
(  <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2) <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z0</font>,2 &amp;  <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c3">f2</font>,2) <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z0</font>,2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="pdiff_3.html#T12" target="_self" onmouseover="rs('pdiff_3/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E4:32"/><span class="kw">then </span>
 <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span></span>)</span>,<font color="Maroon" title="c1">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f1</font>,2)</span>)</span>,<font color="Maroon" title="c1">z0</font>,2)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">f2</font>,2)</span>)</span>,<font color="Maroon" title="c1">z0</font>,2)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T29" onmouseover="rs('pdiff_1/T29')" onmouseout="rh()">PDIFF_1:29</a></span>;<br/>
<span class="kw">then </span> <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span></span>)</span>,<font color="Maroon" title="c1">z0</font>,2) = 
<span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">f2</font>,2)</span>)</span>,<font color="Maroon" title="c1">z0</font>,2)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="pdiff_3.html#T16" target="_self" onmouseover="rs('pdiff_3/T16')" onmouseout="rh()">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="pdiff_3.html#T16" target="_self" onmouseover="rs('pdiff_3/T16')" onmouseout="rh()">Th16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:32"/>
( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z0</font>,2 &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f1</font>,2)</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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">f2</font>,2)</span>)</span></span>)</span>,<font color="Maroon" title="c1">z0</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">z0</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="pdiff_1.html#T29" onmouseover="rs('pdiff_1/T29')" onmouseout="rh()">PDIFF_1:29</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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