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<span class="kw">let </span><font color="Maroon" title="c1">p</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">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> 3</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds   <a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></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> 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">  <a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span><br/>



 <a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T59" title="EUCLID_8:th.59">EUCLID_8:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T59" title="EUCLID_8:th.59">EUCLID_8:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T59" title="EUCLID_8:th.59">EUCLID_8:59</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="pdiff_4.html#E46"><span class="lab"><font color="Green" title="E22">Lm6</font></span></a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="pdiff_4.html#E46"><span class="lab"><font color="Green" title="E22">Lm6</font></span></a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>

;<br/>
<span class="kw">hence </span><a NAME="E2:43"/>
 <a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">p</font>,3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</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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