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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:58"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:58"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> ) )
 
<b>by </b><i><a class="ref" href="seqfunc.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E3:58"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a>  <a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="seqfunc.html#T39" target="_self">Th39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:58_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:58_1"/><b>then </b>
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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/><b>hence </b><a NAME="E4:58_1"/>
<span class="p1">(<span class="default"><a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:58"><i><font color="Green">E41</font></i></a>, <a class="txt" href="seqfunc.html#E1:58_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="seqfunc.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:58"/><i><font color="Green">E43</font></i>: 
 <a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E3:58"><i><font color="Green">E42</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self">Th21</a></i>;<br/>

<i><font color="Green">E44</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:58_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:58_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/><a NAME="E3:58_3"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E4:58_3"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self">Th21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E2:58_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>
<br/>.= 
 <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>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E4:58_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="seq_4.html#T27">SEQ_4:27</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E3:58_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="seqfunc.html#T36" target="_self">Th36</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:58"/>
 <a href="seqfunc.html#K11" target="_self">lim</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K21">abs</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E5:58"><i><font color="Green">E43</font></i></a>, <a class="txt" href="seqfunc.html#E6:58"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>;<br/>

<a NAME="E9:58"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a>  <a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="seqfunc.html#E2:58"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seqfunc.html#T39" target="_self">Th39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:58_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:58_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:58_4"/><b>then </b>
 <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T23">SEQ_2:23</a></i>;<br/><b>hence </b><a NAME="E4:58_4"/>
<span class="p1">(<span class="default"><a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:58"><i><font color="Green">E41</font></i></a>, <a class="txt" href="seqfunc.html#E1:58_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seqfunc.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:58"/><i><font color="Green">E46</font></i>: 
 <a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E9:58"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self">Th21</a></i>;<br/>

<i><font color="Green">E47</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:58_6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:58_6"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/><a NAME="E3:58_6"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E4:58_6"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self">Th21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58_6"><i><font color="Green">E48</font></i></a>, <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E2:58_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E4:58_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="seq_2.html#T24">SEQ_2:24</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E3:58_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seqfunc.html#T36" target="_self">Th36</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:58"/>
 <a href="seqfunc.html#K11" target="_self">lim</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K16">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E11:58"><i><font color="Green">E46</font></i></a>, <a class="txt" href="seqfunc.html#E12:58"><i><font color="Green">E47</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self">Def14</a></i>;<br/>


</div>
