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<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">z0</font>  st <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R3" title="PDIFF_3:pred.3">is_hpartial_differentiable`21_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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">z0</font></span>)</span>,<font color="Olive" title="b1">z0</font></span>)</span> holds <br/>for <font color="Olive" title="b3">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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">V6</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K5" title="RELSET_1:func.5">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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">z0</font></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/><b>let </b><font color="Maroon" title="c2">z0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">z0</font>  st <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R3" title="PDIFF_3:pred.3">is_hpartial_differentiable`21_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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font></span>)</span> holds <br/>for <font color="Olive" title="b2">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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">V6</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K5" title="RELSET_1:func.5">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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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/><b>let </b><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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">z0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R3" title="PDIFF_3:pred.3">is_hpartial_differentiable`21_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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font></span>)</span> implies for <font color="Olive" title="b1">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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">V6</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K5" title="RELSET_1:func.5">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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="pdiff_3.html#R3" title="PDIFF_3:pred.3">is_hpartial_differentiable`21_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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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">V6</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K5" title="RELSET_1:func.5">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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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/>

<b>let </b><font color="Maroon" title="c4">h</font> be  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">c</font> being <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V6</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K5" title="RELSET_1:func.5">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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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/><b>let </b><font color="Maroon" title="c5">c</font> be  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V6</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K5" title="RELSET_1:func.5">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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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/>



<b>assume </b><a NAME="E2:35"/><i><font color="Green" title="E26">A2</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> 1,2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K5" title="RELSET_1:func.5">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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/>

<a NAME="E3:35"/><i><font color="Green" title="E27">A3</font></i>: 
 <a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font> <a href="pdiff_2.html#R1" title="PDIFF_2:pred.1">is_partial_differentiable`1_in</a> <font color="Maroon" title="c2">z0</font>
 
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="pdiff_3.html#T11" target="_self" title="PDIFF_3:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x0</font>, <font color="Maroon" title="c7">y0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:35"/><i><font color="Green" title="E28">A4</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120" title="FINSEQ_2:th.120">FINSEQ_2:120</a></i>;<br/>
 <a href="pdiff_2.html#K3" title="PDIFF_2:func.3">partdiff1</a> <span class="p1">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font> = 
 <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> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c2">z0</font></span>)</span>,<font color="Maroon" title="c6">x0</font>
<b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="#E5:35"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="pdiff_2.html#T11" title="PDIFF_2:th.11">PDIFF_2:11</a></i>
<br/>.= 
 <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font>
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E5:35"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="pdiff_3.html#T7" target="_self" title="PDIFF_3:th.7">Th7</a></i>
;<br/>
<b>hence </b><a NAME="E7:35"/>
( <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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p4">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="pdiff_3.html#K5" title="PDIFF_3:func.5">hpartdiff21</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#K2" title="SEQ_2:func.2">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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> <span class="p5">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z0</font></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> )
 <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E2:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E3:35"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="pdiff_2.html#T15" title="PDIFF_2:th.15">PDIFF_2:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
