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<span class="kw">let </span><font color="Maroon" title="c1">z</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" 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="c1">z</font> holds <br/> <a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</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_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>,1)</span>)</span>,<font color="Maroon" title="c1">z</font>,2)</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> 2</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">z</font> implies  <a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</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_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>,1)</span>)</span>,<font color="Maroon" title="c1">z</font>,2) )</span><br/>



<a NAME="E1:26"/>
<font color="Maroon" title="c1">z</font> is   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 2, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T131" title="FINSEQ_2:th.131">FINSEQ_2:131</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T100" title="FINSEQ_2:th.100">FINSEQ_2:100</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:26"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</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_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>,1)</span>)</span>,<font color="Maroon" title="c1">z</font>,2)</span><br/>

<a NAME="E5:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c2">f</font>,1) <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z</font>,2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_3.html#T10" target="_self" title="PDIFF_3:th.10">Th10</a></span>;<br/>
 <a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</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#K1" title="PDIFF_2:func.1">SVF1</a> (2,<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>,1)</span>)</span>,<font color="Maroon" title="c1">z</font>)</span>)</span>,<font color="Maroon" title="c4">y0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="pdiff_3.html#T6" target="_self" title="PDIFF_3:th.6">Th6</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>,1)</span>)</span>,<font color="Maroon" title="c1">z</font>,2)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="pdiff_2.html#T14" title="PDIFF_2:th.14">PDIFF_2:14</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:26"/>
 <a href="pdiff_3.html#K3" title="PDIFF_3:func.3">hpartdiff12</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_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>,1)</span>)</span>,<font color="Maroon" title="c1">z</font>,2)
 ; <a class="txt" 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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