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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green" title="E41">A2</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> ) )
 
<b>by </b><i><a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/>
<a NAME="E3:57"/>
( <font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c3">H2</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> ) )
 
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font> &amp; <font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> &amp; <font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> )
 
<b>by </b><i><a class="txt" href="seqfunc.html#E2:57"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="seqfunc.html#T38" target="_self" title="SEQFUNC:th.38">Th38</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" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><a NAME="E2:57_1"/><b>then </b><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E3:57_1"/>
<font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E1:57_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E4:57_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</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="seqfunc.html#E2:57_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/><b>hence </b><a NAME="E5:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E1:57_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:57"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E4:57"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/>

<i><font color="Green" title="E44">A7</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">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" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>assume </b><a NAME="E1:57_3"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:57_3"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/><a NAME="E3:57_3"/><b>then </b><i><font color="Green" title="E46">A9</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:57_3"/><b>then </b><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>;<br/><a NAME="E5:57_3"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E6:57_3"/><i><font color="Green" title="E49">A12</font></i>: 
<font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E4:57_3"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57_3"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
<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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E3:57_3"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E3:57_3"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E5:57_3"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="seqfunc.html#E6:57_3"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E4:57_3"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:57"/>
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E6:57"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="seqfunc.html#E7:57"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</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" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>assume </b><a NAME="E1:57_4"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><a NAME="E2:57_4"/><b>then </b><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E3:57_4"/>
<font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E1:57_4"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E4:57_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</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="seqfunc.html#E2:57_4"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/><b>hence </b><a NAME="E5:57_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E1:57_4"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:57"/><i><font color="Green" title="E45">A15</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E4:57"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/>

<i><font color="Green" title="E46">A16</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">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" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>assume </b><a NAME="E1:57_6"/><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:57_6"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>;<br/><a NAME="E3:57_6"/><b>then </b><i><font color="Green" title="E48">A18</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:57_6"/><b>then </b><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>;<br/><a NAME="E5:57_6"/><b>then </b><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E6:57_6"/><i><font color="Green" title="E51">A21</font></i>: 
<font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E4:57_6"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57_6"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E3:57_6"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E3:57_6"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E5:57_6"><i><font color="Green" title="E50">A20</font></i></a>, <a class="txt" href="seqfunc.html#E6:57_6"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E4:57_6"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:57"/>
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E11:57"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="seqfunc.html#E12:57"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</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" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>assume </b><a NAME="E1:57_7"/><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><a NAME="E2:57_7"/><b>then </b><i><font color="Green" title="E48">A23</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E3:57_7"/>
<font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E1:57_7"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E4:57_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</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="seqfunc.html#E2:57_7"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/><b>hence </b><a NAME="E5:57_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E1:57_7"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:57"/><i><font color="Green" title="E47">A24</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E4:57"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/>

<i><font color="Green" title="E48">A25</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">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" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>assume </b><a NAME="E1:57_9"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:57_9"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></i>;<br/><a NAME="E3:57_9"/><b>then </b><i><font color="Green" title="E49">A26</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:57_9"/><b>then </b><i><font color="Green" title="E50">A27</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>;<br/><a NAME="E5:57_9"/><b>then </b><i><font color="Green" title="E51">A28</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><a NAME="E6:57_9"/><i><font color="Green" title="E52">A29</font></i>: 
<font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E4:57_9"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<br/>.= 
<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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E3:57_9"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
<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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E3:57_9"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E5:57_9"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="seqfunc.html#E6:57_9"><i><font color="Green" title="E52">A29</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E4:57_9"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:57"/>
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E16:57"><i><font color="Green" title="E47">A24</font></i></a>, <a class="txt" href="seqfunc.html#E17:57"><i><font color="Green" title="E48">A25</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>;<br/>


</div>
