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<b>let </b><font color="Maroon" title="c1">z</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>  holds <br/> ( <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> iff  <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">z</font> <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><br/><b>let </b><font color="Maroon" title="c2">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="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> iff  <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <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><br/>



<b>thus </b><a NAME="E1:26"/>
( <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#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z</font>,2 )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z</font>,2 implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <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><br/>

<a NAME="E2:26_1"/><b>then </b>
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="ref" href="pdiff_3.html#T10" target="_self" title="PDIFF_3:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1"/>
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z</font>,2
 <b>by </b><i><a class="ref" href="pdiff_2.html#T8" title="PDIFF_2:th.8">PDIFF_2:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:26"/>
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">z</font>,2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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></span><br/>

<a NAME="E3:26"/><b>then </b>
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">z</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c1">z</font>
 
<b>by </b><i><a class="ref" href="pdiff_2.html#T8" title="PDIFF_2:th.8">PDIFF_2:8</a></i>;<br/>
<b>hence </b><a NAME="E4:26"/>
<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>
 <b>by </b><i><a class="ref" href="pdiff_3.html#T10" target="_self" title="PDIFF_3:th.10">Th10</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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