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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:43"/>
<font color="Maroon">c<sub>4</sub></font> <a href="fdiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:43"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="fdiff_1.html#NM2">LINEAR</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="fdiff_1.html#NM1">REST</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="fdiff_1.html#NM2">LINEAR</a>, <font color="Maroon">c<sub>7</sub></font> being   <a href="fdiff_1.html#NM1">REST</a><b> such that </b><br/><a NAME="E5:43"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:43"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E8:43"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a></i>;<br/>
<a NAME="E9:43"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E5:43"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E10:43"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 0</span>)</span>
 
;<br/>
<a NAME="E11:43"/><b>then </b><i><font color="Green">E28</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E7:43"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:43"/>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E13:43"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="partfun1.html#V1">total</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="fdiff_1.html#V1" target="_self">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 ) ) )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="fdiff_1.html#V1" target="_self">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><a NAME="E1:43_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E2:43_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T27">SEQ_2:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:43_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> ) )
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E5:43_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/><a NAME="E6:43_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E4"><i><font color="Green">Lemma3</font></i></a></i>;<br/><a NAME="E7:43_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/><a NAME="E8:43_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E9:43_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T26">SEQ_4:26</a></i>;<br/><i><font color="Green">E37</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E8:43_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_4.html#T27">SEQ_4:27</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
;<br/><a NAME="E11:43_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E9:43_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/><i><font color="Green">E39</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> 0
<b>by </b><i><a class="txt" href="fdiff_1.html#E9:43_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="fdiff_1.html#E10:43_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="fdiff_1.html#E6:43_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>
;<br/><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:43_1_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E8:43_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
;<br/><a NAME="E3:43_1_3"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E4:43_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E4:43_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E5:43_1_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/><b>hence </b><a NAME="E6:43_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E4"><i><font color="Green">Lemma3</font></i></a></i>;<br/><a NAME="E7:43_1_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E8:43_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E3:43_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:43_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E10:43_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E11:43_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:43_1_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E12:43_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/><a NAME="E13:43_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E14:43_1_3"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/></div><b>end;</b></div><a NAME="E14:43_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E5:43_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="fdiff_1.html#E11:43_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="fdiff_1.html#E12:43_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seq_2.html#T33">SEQ_2:33</a></i>;<br/><a NAME="E15:43_1"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E5:43_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="fdiff_1.html#E7:43_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="fdiff_1.html#E11:43_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="fdiff_1.html#E12:43_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="fdiff_1.html#E13:43_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seq_2.html#T34">SEQ_2:34</a></i>;<br/><b>hence </b><a NAME="E16:43_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 0 )
 <b>by </b><i><a class="txt" href="fdiff_1.html#E11:43"><i><font color="Green">E28</font></i></a>, <a class="txt" href="fdiff_1.html#E14:43_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seq_4.html#T28">SEQ_4:28</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 ;<br/><a NAME="E2:43_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><a NAME="E3:43_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is  <a href="fdiff_1.html#V1" target="_self">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>
 <b>by </b><i><a class="txt" href="fdiff_1.html#E1:43_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="fdiff_1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:43_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 0 )
 <b>by </b><i><a class="txt" href="fdiff_1.html#E14:43"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:43"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="fdiff_1.html#E13:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:43"/>
<font color="Maroon">c<sub>7</sub></font> <a href="fcont_1.html#R1">is_continuous_in</a> 0
 <b>by </b><i><a class="txt" href="fdiff_1.html#E15:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fcont_1.html#T2">FCONT_1:2</a></i>;<br/>


</div>
