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<b>let </b><font color="Maroon" title="c1">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"> for <font color="Olive" title="b1">z</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 holds <br/> ( <font color="Maroon" title="c1">f</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Olive" title="b1">z</font> iff <font color="Maroon" title="c1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">z</font>,1 )</span><br/><b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z</font> iff <font color="Maroon" title="c1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z</font>,1 )</span><br/>



<b>thus </b><a NAME="E1:11"/>
( <font color="Maroon" title="c1">f</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z</font> implies <font color="Maroon" title="c1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z</font>,1 )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z</font>,1 implies <font color="Maroon" title="c1">f</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E10">A0</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z</font>,1</span><br/>

<b>consider </b><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><b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c2">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> &amp;  <a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font> )
 <b>by </b><i><a class="txt" href="#E1:11_1"><i><font color="Green" title="E10">A0</font></i></a>, <a class="ref" href="pdiff_2.html#D3" target="_self" title="PDIFF_2:def.3">Def6</a></i>;<br/>
<a NAME="E4:11_1"/><i><font color="Green" title="E12">AA</font></i>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="#E3:11_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="pdiff_2.html#T1" target="_self" title="PDIFF_2:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E5:11_1"/>
<font color="Maroon" title="c1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z</font>,1
 <b>by </b><i><a class="txt" href="#E3:11_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E4:11_1"><i><font color="Green" title="E12">AA</font></i></a>, <a class="ref" href="pdiff_1.html#D11" title="PDIFF_1:def.11">PDIFF_1:def 11</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="E2:11"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z</font>,1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z</font></span><br/>

<b>consider </b><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><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c2">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>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120" title="FINSEQ_2:th.120">FINSEQ_2:120</a></i>;<br/>
<a NAME="E5:11"/><i><font color="Green" title="E12">A4</font></i>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="pdiff_2.html#T1" target="_self" title="PDIFF_2:th.1">Th1</a></i>;<br/>
<a NAME="E6:11"/>
 <a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="#E2:11"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E5:11"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="pdiff_1.html#D11" title="PDIFF_1:def.11">PDIFF_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E7:11"/>
<font color="Maroon" title="c1">f</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z</font>
 <b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="pdiff_2.html#D3" target="_self" title="PDIFF_2:def.3">Def6</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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