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<span class="kw">let </span><font color="Maroon" title="c1">u0</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> 3; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</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> 3</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f1</font> <a href="pdiff_5.html#R2" title="PDIFF_5:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">u0</font> &amp; <font color="Olive" title="b2">f2</font> <a href="pdiff_5.html#R2" title="PDIFF_5:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">u0</font> holds <br/>( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b1">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b2">f2</font>,1</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,2 &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b1">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Olive" title="b2">f2</font>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">u0</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c1">u0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">u0</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</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> 3</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="pdiff_5.html#R2" title="PDIFF_5:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">u0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="pdiff_5.html#R2" title="PDIFF_5:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">u0</font> implies ( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,2 &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">u0</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">u0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">u0</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="pdiff_5.html#R2" title="PDIFF_5:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">u0</font>
 <span class="kw">and </span><br/><a NAME="E2:67"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c3">f2</font> <a href="pdiff_5.html#R2" title="PDIFF_5:pred.2">is_hpartial_differentiable`12_in</a> <font color="Maroon" title="c1">u0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,2 &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">u0</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">u0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">u0</font></span>)</span> )</span><br/>

<a NAME="E3:67"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1 <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="pdiff_5.html#T20" target="_self" title="PDIFF_5:th.20">Th10</a></span>;<br/>
<a NAME="E4:67"/><span class="lab"><font color="Green" title="E58">AA</font></span>: 
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1 <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="pdiff_5.html#T20" target="_self" title="PDIFF_5:th.20">Th10</a></span>;<br/>
<a NAME="E5:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A4</font></span>: 
( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,2 &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">u0</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span>,<font color="Maroon" title="c1">u0</font>,2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span>,<font color="Maroon" title="c1">u0</font>,2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="pdiff_1.html#T29" title="PDIFF_1:th.29">PDIFF_1:29</a></span>;<br/>
 <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">u0</font>,2 = 
<span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">u0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span>,<font color="Maroon" title="c1">u0</font>,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="pdiff_5.html#T29" target="_self" title="PDIFF_5:th.29">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">u0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">u0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="pdiff_5.html#T29" target="_self" title="PDIFF_5:th.29">Th18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:67"/>
( <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c1">u0</font>,2 &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c2">f1</font>,1</span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> <font color="Maroon" title="c3">f2</font>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">u0</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">u0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pdiff_5.html#K2" title="PDIFF_5:func.2">hpartdiff12</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">u0</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E58">AA</font></span></a>, <a class="ref" href="pdiff_1.html#T29" title="PDIFF_1:th.29">PDIFF_1:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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