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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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>   st <font color="Olive" title="b1">f</font> <a href="pdiff_4.html#R2" title="PDIFF_4:pred.2">is_partial_differentiable`2_on</a> <font color="Maroon" title="c1">D</font> holds <br/>( <font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; ( for <font color="Olive" title="b2">u</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> 3  st <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Olive" title="b1">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b2">u</font>,2 ) )</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"> ( <font color="Maroon" title="c2">f</font> <a href="pdiff_4.html#R2" title="PDIFF_4:pred.2">is_partial_differentiable`2_on</a> <font color="Maroon" title="c1">D</font> implies ( <font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; ( for <font color="Olive" title="b1">u</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> 3  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">u</font>,2 ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="pdiff_4.html#R2" title="PDIFF_4:pred.2">is_partial_differentiable`2_on</a> <font color="Maroon" title="c1">D</font>
 ; <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="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; ( for <font color="Olive" title="b1">u</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> 3  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">u</font>,2 ) )</span><br/>

<span class="kw">hence </span><a NAME="E2:24"/>
<font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_4.html#D2" target="_self" title="PDIFF_4:def.2">Def17</a></span>; <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">u</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> 3  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">u</font>,2</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">g</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">u0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> implies <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c4">u0</font>,2 )</span><br/>

<span class="kw">assume </span><a NAME="E3:24"/>
<font color="Maroon" title="c4">u0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 ; <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">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c4">u0</font>,2</span><br/>

<a NAME="E4:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c4">u0</font>,2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="pdiff_4.html#D2" target="_self" title="PDIFF_4:def.2">Def17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">x0</font>, <font color="Maroon" title="c6">y0</font>, <font color="Maroon" title="c7">z0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:24"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
( <font color="Maroon" title="c4">u0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">x0</font>,<font color="Maroon" title="c6">y0</font>,<font color="Maroon" title="c7">z0</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp;  ex <font color="Olive" title="b1">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c6">y0</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> &amp;  ex <font color="Olive" title="b2">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b3">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>for <font color="Olive" title="b4">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="pdiff_4.html#T14" target="_self" title="PDIFF_4:th.14">BXXLXSDef11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c6">y0</font><span class="kw"> such that </span><br/><a NAME="E8:24"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
( <font color="Maroon" title="c8">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> &amp;  ex <font color="Olive" title="b1">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b2">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>for <font color="Olive" title="b3">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/>
<a NAME="E9:24"/>
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span>
 ; <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="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span></span><br/>

<a NAME="E2:24_1"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> 2,<font color="Maroon" title="c4">u0</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> 2,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/>
<a NAME="E3:24_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E4:24_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:24_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</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><span class="kw">end;</span></div>
<a NAME="E10:24"/><span class="kw">then </span>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span>
 
;<br/>
<a NAME="E11:24"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E12:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c8">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">L</font> being   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a>, <font color="Maroon" title="c10">R</font> being   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a><span class="kw"> such that </span><br/><a NAME="E14:24"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>;<br/>
<a NAME="E15:24"/>
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_2"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</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"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:24_2"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> implies <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_2_1"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span>
 ; <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_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font></span><br/>

<a NAME="E2:24_2_1"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> 2,<font color="Maroon" title="c4">u0</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> 2,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/>
<span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> 2,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> 2,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:24_2_1"/>
<span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 ; <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><span class="kw">end;</span></div>
<a NAME="E3:24_2"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c6">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T37" title="RCOMP_1:th.37">RCOMP_1:37</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:24_2"><span class="lab"><font color="Green" title="E28">A10</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:24_2"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E3:24_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:24_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> 2,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">u0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:24"/>
<font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c4">u0</font>,2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="pdiff_4.html#T14" target="_self" title="PDIFF_4:th.14">BXXLXSDef11</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>
