<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">a</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#K6" title="PDIFF_4:func.6">Directiondiff</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><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></span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_4.html#K5" title="PDIFF_4:func.5">unitvector</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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#K6" title="PDIFF_4:func.6">Directiondiff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_4.html#K5" title="PDIFF_4:func.5">unitvector</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">p0</font> =  <a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">q0</font> =  <a href="pdiff_4.html#K5" title="PDIFF_4:func.5">unitvector</a> <font color="Maroon" title="c2">a</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">g1</font> =  <a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c7">g2</font> =  <a href="pdiff_4.html#K5" title="PDIFF_4:func.5">unitvector</a> <font color="Maroon" title="c2">a</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="lab"><font color="Green" title="E27">A1</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T1" title="EUCLID_8:th.1">EUCLID_8:1</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
;<br/>
<a NAME="E3:53"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E4:53"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">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="c3">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="c3">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="c3">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="ref" href="pdiff_4.html#T37" target="_self" title="PDIFF_4:th.37">Th43</a></span>;<br/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_4.html#K5" title="PDIFF_4:func.5">unitvector</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="euclid_5.html#T28" title="EUCLID_5:th.28">EUCLID_5:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T108" title="RVSUM_1:th.108">RVSUM_1:108</a></span>
<br/>.= 
<span class="p1">(<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="c3">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
<br/>.= 
<span class="p1">(<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="c3">f</font>,<font color="Maroon" title="c1">p</font>,1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1: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="c3">f</font>,<font color="Maroon" title="c1">p</font>,2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
<br/>.= 
 <a href="pdiff_4.html#K6" title="PDIFF_4:func.6">Directiondiff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:53"/>
 <a href="pdiff_4.html#K6" title="PDIFF_4:func.6">Directiondiff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_4.html#K4" title="PDIFF_4:func.4">grad</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="pdiff_4.html#K5" title="PDIFF_4:func.5">unitvector</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</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/>


</div>
