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

<span class="kw">let </span><font color="Maroon" title="c1">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> 2</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z0</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> 2<br/>  for <font color="Olive" title="b2">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z0</font>  st <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Olive" title="b1">z0</font> &amp; <font color="Olive" title="b2">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"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Olive" title="b1">z0</font>)</span>)</span> holds <br/> for <font color="Olive" title="b3">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b4">c</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Olive" title="b1">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Olive" title="b1">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b4">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Olive" title="b1">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Olive" title="b1">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b4">c</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">z0</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> 2; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font>  st <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c2">z0</font> &amp; <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"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> holds <br/> for <font color="Olive" title="b2">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b3">c</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font> be    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c2">z0</font> &amp; <font color="Maroon" title="c3">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"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> implies  for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">c</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R4" title="PDIFF_3:pred.4">is_hpartial_differentiable`22_in</a> <font color="Maroon" title="c2">z0</font>
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c3">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"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">c</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">c</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">c</font> be   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span></span>)</span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E3:28"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">z0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">x0</font>, <font color="Maroon" title="c7">y0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E5:28"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c2">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c6">x0</font>,<font color="Maroon" title="c7">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T100" onmouseover="rs('finseq_2/T100')" onmouseout="rh()">FINSEQ_2:100</a></span>;<br/>
<a NAME="E6:28"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 <a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2) <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c2">z0</font>,2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="pdiff_3.html#T12" target="_self" onmouseover="rs('pdiff_3/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">then </span> <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>,2) = 
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</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"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span>,<font color="Maroon" title="c7">y0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="pdiff_2.html#T14" onmouseover="rs('pdiff_2/T14')" onmouseout="rh()">PDIFF_2:14</a></span>
<br/>.= 
 <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="pdiff_3.html#T8" target="_self" onmouseover="rs('pdiff_3/T8')" onmouseout="rh()">Th8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:28"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p4">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff22</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> (2,<span class="p5">(<span class="default"><a href="pdiff_3.html#K1" title="PDIFF_3:func.1">pdiff1</a> (<font color="Maroon" title="c1">f</font>,2)</span>)</span>,<font color="Maroon" title="c2">z0</font>)</span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="pdiff_2.html#T18" onmouseover="rs('pdiff_2/T18')" onmouseout="rh()">PDIFF_2:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
