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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" 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#R1" title="PDIFF_3:pred.1">is_hpartial_differentiable`11_in</a> <font color="Maroon" title="c1">z</font> holds <br/> <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">hpartdiff11</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#K10" title="PDIFF_1:func.10">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>,1)</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" 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">f</font> <a href="pdiff_3.html#R1" title="PDIFF_3:pred.1">is_hpartial_differentiable`11_in</a> <font color="Maroon" title="c1">z</font> implies  <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">hpartdiff11</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#K10" title="PDIFF_1:func.10">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>,1) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">x0</font>, <font color="Maroon" title="c4">y0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E14">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" onmouseover="rs('finseq_2/T100')" onmouseout="rh()">FINSEQ_2:100</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:21"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="pdiff_3.html#R1" title="PDIFF_3:pred.1">is_hpartial_differentiable`11_in</a> <font color="Maroon" title="c1">z</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">  <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">hpartdiff11</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#K10" title="PDIFF_1:func.10">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>,1)</span><br/>

<a NAME="E4:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">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>,1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_3.html#T9" target="_self" onmouseover="rs('pdiff_3/T9')" onmouseout="rh()">Th9</a></span>;<br/>
 <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">hpartdiff11</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> (1,<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="c3">x0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="pdiff_3.html#T5" target="_self" onmouseover="rs('pdiff_3/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
 <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">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>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="pdiff_2.html#T13" onmouseover="rs('pdiff_2/T13')" onmouseout="rh()">PDIFF_2:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:21"/>
 <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">hpartdiff11</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#K10" title="PDIFF_1:func.10">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>,1)
 ; <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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