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<div class="add">

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">x0</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font><b> such that </b><br/><a NAME="E3:40"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">f</font> &amp;  ex <font color="Olive" title="b1">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b2">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>for <font color="Olive" title="b3">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D5" target="_self" title="FDIFF_1:def.5">Def5</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T37" title="RCOMP_1:th.37">RCOMP_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E28">A4</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</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> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c6">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c7">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E29">A5</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7" title="RCOMP_1:def.7">RCOMP_1:def 7</a></i>;<br/><a NAME="E4:40_1"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">s2</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c4">x0</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/><a NAME="E6:40_1"/><i><font color="Green" title="E30">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>) <b>-&gt; </b>   <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="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c9">s3</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E8:40_1"/><i><font color="Green" title="E31">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i><a NAME="E9:40_1"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c8">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/><i><font color="Green" title="E33">A9</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s2</font> = 
<font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
<font color="Maroon" title="c4">x0</font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E6:40_1"><i><font color="Green" title="E30">A6</font></i></a></i>
;<br/><a NAME="E11:40_1"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c9">s3</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E8:40_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>;<br/><a NAME="E12:40_1"/><b>then </b><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c9">s3</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fdiff_1.html#E9:40_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/><i><font color="Green" title="E36">A12</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">s3</font> = 
<font color="Maroon" title="c4">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x0</font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fdiff_1.html#E9:40_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="fdiff_1.html#E10:40_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="fdiff_1.html#E11:40_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>
<br/>.= 
0

;<br/><div><i><font color="Green" title="E37">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E1:40_1_3"/><i><font color="Green" title="E38">A14</font></i>: 
<font color="Maroon" title="c9">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:40_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E8:40_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="fdiff_1.html#E1:40_1_3"><i><font color="Green" title="E38">A14</font></i></a></i>;<br/><a NAME="E3:40_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E6:40_1"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/><a NAME="E4:40_1_3"/><b>then </b>
<font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 ;<br/><b>hence </b><a NAME="E5:40_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c10">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E16:40_1"/><i><font color="Green" title="E38">A15</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fdiff_1.html#E3:40_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/><a NAME="E17:40_1"/><i><font color="Green" title="E39">A16</font></i>: 
( <font color="Maroon" title="c9">s3</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s3</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="fdiff_1.html#E12:40_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="fdiff_1.html#E13:40_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>;<br/><div><i><font color="Green" title="E40">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E1:40_1_4"/><i><font color="Green" title="E41">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s3</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:40_1_4"/>
<font color="Maroon" title="c9">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1_4"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/><b>hence </b><a NAME="E3:40_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_1.html#E14:40_1"><i><font color="Green" title="E37">A13</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E19:40_1"/><b>then </b>
<font color="Maroon" title="c9">s3</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c9">s3</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font> as  <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> <b>by </b><i><a class="txt" href="fdiff_1.html#E17:40_1"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="fdiff_1.html#D1" target="_self" title="FDIFF_1:def.1">Def1</a></i>;<br/><a NAME="E21:40_1"/><i><font color="Green" title="E41">A19</font></i>: 
<font color="Maroon" title="c8">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T47" title="SEQM_3:th.47">SEQM_3:47</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c8">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font> as  <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> <b>by </b><i><a class="txt" href="fdiff_1.html#E21:40_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="seqm_3.html#T54" title="SEQM_3:th.54">SEQM_3:54</a></i>;<br/><a NAME="E23:40_1"/><i><font color="Green" title="E42">A20</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">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"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_1_5"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_5_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">c</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_5_1"/><i><font color="Green" title="E43">A21</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:40_1_5_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E3:40_1_5_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E5:40_1_5_1"/><b>then </b>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E6:40_1"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:40_1_5_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:40_1_5"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E3:40_1_5"/><b>then </b><i><font color="Green" title="E43">A22</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<font color="Maroon" title="c12">c</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
<font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E6:40_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="fdiff_1.html#E3:40_1_5"><i><font color="Green" title="E43">A22</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:40_1_5"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">c</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:40_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_6"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:40_1_6"/><i><font color="Green" title="E43">A23</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<i><font color="Green" title="E44">A24</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E11:40_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="seqm_3.html#T37" title="SEQM_3:th.37">SEQM_3:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">n</font></span>)</span>

;<br/>
<a NAME="E5:40_1_6"/>
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E6:40_1_6"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E16:40_1"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="fdiff_1.html#E4:40_1_6"><i><font color="Green" title="E44">A24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_6"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:40_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="fdiff_1.html#E3:40_1_6"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:40_1"/><b>then </b><i><font color="Green" title="E43">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:40"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="fdiff_1.html#E3:40"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="fdiff_1.html#E23:40_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="fdiff_1.html#T20" target="_self" title="FDIFF_1:th.20">Th20</a></i>;<br/><b>then </b><i><font color="Green" title="E44">A26</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E17:40_1"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
0

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:40_1_8"/><i><font color="Green" title="E45">A27</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E18:40_1"><i><font color="Green" title="E40">A17</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8" title="SEQ_1:def.8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span>

<br/>.= 
1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1_8"><i><font color="Green" title="E45">A27</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font>

;<br/></div><b>end;</b></div><a NAME="E28:40_1"/><b>then </b><i><font color="Green" title="E45">A28</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><a NAME="E29:40_1"/><b>then </b><i><font color="Green" title="E46">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E17:40_1"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="fdiff_1.html#E25:40_1"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/><div><i><font color="Green" title="E47">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:40_1_9"/><i><font color="Green" title="E48">A31</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">p</font>
 ;<br/><b>take </b><font color="Maroon" title="c14">n</font> = 0;<br/><b>let </b><font color="Maroon" title="c15">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E2:40_1_9"/>
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ;<br/><a NAME="E3:40_1_9"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T37" title="RCOMP_1:th.37">RCOMP_1:37</a></i>;<br/><a NAME="E4:40_1_9"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">c</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> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:40"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="fdiff_1.html#E23:40_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>then </b> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E6:40_1"><i><font color="Green" title="E30">A6</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
;<br/><b>hence </b><a NAME="E6:40_1_9"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">p</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1_9"><i><font color="Green" title="E48">A31</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E31:40_1"/><b>then </b><i><font color="Green" title="E48">A32</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/><a NAME="E32:40_1"/><b>then </b><i><font color="Green" title="E49">A33</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E30:40_1"><i><font color="Green" title="E47">A30</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/><a NAME="E33:40_1"/><i><font color="Green" title="E50">A34</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E29:40_1"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="fdiff_1.html#E31:40_1"><i><font color="Green" title="E48">A32</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/><i><font color="Green" title="E51">A35</font></i>:  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> = 
0 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E26:40_1"><i><font color="Green" title="E44">A26</font></i></a>, <a class="txt" href="fdiff_1.html#E28:40_1"><i><font color="Green" title="E45">A28</font></i></a>, <a class="txt" href="fdiff_1.html#E29:40_1"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="fdiff_1.html#E31:40_1"><i><font color="Green" title="E48">A32</font></i></a>, <a class="txt" href="fdiff_1.html#E32:40_1"><i><font color="Green" title="E49">A33</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</font>

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font>

;<br/></div><b>end;</b></div><a NAME="E36:40_1"/><b>then </b><i><font color="Green" title="E52">A36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E11:40_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="seqm_3.html#T37" title="SEQM_3:th.37">SEQM_3:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
;<br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E53">A37</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c12">c</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_1.html#E36:40_1"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c10">l</font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E1:40_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="rfunct_2.html#T22" title="RFUNCT_2:th.22">RFUNCT_2:22</a></i>
;<br/><b>hence </b><a NAME="E39:40_1"/>
<font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E33:40_1"><i><font color="Green" title="E50">A34</font></i></a>, <a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a></i>;<br/><b>thus </b><a NAME="E40:40_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x0</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"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E33:40_1"><i><font color="Green" title="E50">A34</font></i></a>, <a class="txt" href="fdiff_1.html#E34:40_1"><i><font color="Green" title="E51">A35</font></i></a>, <a class="txt" href="fdiff_1.html#E38:40_1"><i><font color="Green" title="E53">A37</font></i></a>, <a class="ref" href="seq_4.html#T36" title="SEQ_4:th.36">SEQ_4:36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:40"/>
<font color="Maroon" title="c3">f</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:40"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="fdiff_1.html#E4:40"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="fcont_1.html#T2" title="FCONT_1:th.2">FCONT_1:2</a></i>;<br/>


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