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

<b>let </b><font color="Maroon">D</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">H1</font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">H2</font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">H1</font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">H2</font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">X</font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">H1</font> &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Olive">x</font> is <a href="seq_2.html#V4">convergent</a> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:57"/>
( <font color="Maroon">X</font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">H2</font> &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Olive">x</font> is <a href="seq_2.html#V4">convergent</a> ) )
 
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">X</font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">H1</font> <a href="seqfunc.html#K6" target="_self">+</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">X</font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">H1</font> <a href="seqfunc.html#K7" target="_self">-</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">X</font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">H1</font> <a href="seqfunc.html#K8" target="_self">(#)</a> <font color="Maroon">H2</font> )
 
<b>by </b><i><a class="txt" href="seqfunc.html#E4:57"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:57_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E3:57_1"/>
<font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E2:57_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E4:57_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E6:57"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seq_2.html#T19">SEQ_2:19</a></i>;<br/><b>hence </b><a NAME="E5:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K6" target="_self">+</a> <font color="Maroon">H2</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E2:57_1"><i><font color="Green">E43</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:57"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K6" target="_self">+</a> <font color="Maroon">H2</font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<i><font color="Green">E72</font></i>:  <a href="relset_1.html#K4">dom</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">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">X</font>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:57_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:57_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/><a NAME="E3:57_3"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</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">H1</font>,<font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">x</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">H2</font>,<font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:57_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E5:57_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E6:57_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</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">H2</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E4"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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">H2</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K6" target="_self">+</a> <font color="Maroon">H2</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:57"/>
 <a href="seqfunc.html#K11" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K6" target="_self">+</a> <font color="Maroon">H2</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="seqfunc.html#E6:57"><i><font color="Green">E44</font></i></a>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:57_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:57_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E3:57_4"/>
<font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E4"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/><a NAME="E4:57_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seqfunc.html#T2" target="_self">Th2</a>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/><b>hence </b><a NAME="E5:57_4"/>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K7" target="_self">-</a> <font color="Maroon">H2</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="txt" href="seqfunc.html#E4"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:57"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K7" target="_self">-</a> <font color="Maroon">H2</font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<i><font color="Green">E103</font></i>:  <a href="relset_1.html#K4">dom</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">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">X</font>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:57_6"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:57_6"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/><a NAME="E3:57_6"/><b>then </b><i><font color="Green">E105</font></i>: 
( <font color="Maroon">x</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">H1</font>,<font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">x</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">H2</font>,<font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:57_6"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E5:57_6"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E6:57_6"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</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">H2</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">H2</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D2" target="_self">Def2</a>, , <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K7" target="_self">-</a> <font color="Maroon">H2</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:57"/>
 <a href="seqfunc.html#K11" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K7" target="_self">-</a> <font color="Maroon">H2</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E4"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="seqfunc.html#T2" target="_self">Th2</a>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:57_7"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:57_7"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E3:57_7"/>
<font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>;<br/><a NAME="E4:57_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/><b>hence </b><a NAME="E5:57_7"/>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K8" target="_self">(#)</a> <font color="Maroon">H2</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:57"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K8" target="_self">(#)</a> <font color="Maroon">H2</font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<i><font color="Green">E112</font></i>:  <a href="relset_1.html#K4">dom</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">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">X</font>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:57_9"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:57_9"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><a NAME="E3:57_9"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">x</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">H1</font>,<font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">x</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">H2</font>,<font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:57_9"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E5:57_9"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E6:57_9"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</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">H2</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">H2</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D3" target="_self">Def3</a>, <a class="ref" href="seqfunc.html#D4" target="_self">Def4</a>, <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K8" target="_self">(#)</a> <font color="Maroon">H2</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green">E40</font></i></a>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:57"/>
 <a href="seqfunc.html#K11" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="seqfunc.html#K8" target="_self">(#)</a> <font color="Maroon">H2</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H1</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H2</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
