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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">z0</font> being   <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  st <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_in</a> <font color="Olive" title="b1">z0</font> holds <br/> <a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">z0</font></span>)</span>,<font color="Olive" title="b1">z0</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 2,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">z0</font></span><br/><b>let </b><font color="Maroon" title="c2">z0</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"> ( <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c2">z0</font> implies  <a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 2,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">z0</font> )</span><br/>



<b>assume </b><a NAME="E1:54"/>
<font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R2" title="PDIFF_3:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c2">z0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 2,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">z0</font></span><br/>

<a NAME="E2:54"/><b>then </b>
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font> <a href="pdiff_2.html#R2" title="PDIFF_2:pred.2">is_partial_differentiable`2_in</a> <font color="Maroon" title="c2">z0</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="E3:54"/>
 <a href="pdiff_2.html#K2" title="PDIFF_2:func.2">SVF2</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 2,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">z0</font>
 <b>by </b><i><a class="ref" href="pdiff_2.html#T26" title="PDIFF_2:th.26">PDIFF_2:26</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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