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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">h</font> be   <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" 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="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )</span><br/><span class="lab"><font color="Green" title="E7">A1</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T15" title="CFCONT_1:th.15">CFCONT_1:15</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_1.html#T13" title="COMSEQ_1:th.13">COMSEQ_1:13</a></span>
;<br/><a NAME="E2:22_1_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D3" target="_self" title="CFDIFF_1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:22_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span><br/><a NAME="E4:22_1_1_1"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D3" target="_self" title="CFDIFF_1:def.3">Def3</a></span>;<br/><span class="kw">hence </span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:22_1_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="comseq_2.html#T18" title="COMSEQ_2:th.18">COMSEQ_2:18</a></span>
<br/>.= 
 <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:22_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">R</font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="cfdiff_1.html#V2" title="CFDIFF_1:attr.2">REST-like</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D3" target="_self" title="CFDIFF_1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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