<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">H</font> being   <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/> for <font color="Olive" title="b3">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b3">X</font> holds <br/>( <font color="Olive" title="b1">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Olive" title="b2">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b3">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Olive" title="b2">H</font></span>)</span>,<font color="Olive" title="b3">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b2">H</font>,<font color="Olive" title="b3">X</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <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/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b2">X</font> holds <br/>( <font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Olive" title="b1">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b2">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Olive" title="b2">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b1">H</font>,<font color="Olive" title="b2">X</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">H</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c3">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b1">X</font> holds <br/>( <font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b1">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Olive" title="b1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Olive" title="b1">X</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> implies ( <font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span> ) )</span><br/>







<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

<a NAME="E2:59"/><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="c3">H</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">H</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="E3:59"/><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">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="ref" href="seqfunc.html#T40" target="_self" title="SEQFUNC:th.40">Th40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</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> )</span><br/><b>assume </b><a NAME="E1:59_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</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></span><br/><a NAME="E2:59_1"/><b>then </b>
<font color="Maroon" title="c3">H</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#E1:59"><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="E3:59_1"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</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="ref" href="seq_2.html#T21" title="SEQ_2:th.21">SEQ_2:21</a></i>;<br/><b>hence </b><a NAME="E4:59_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</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#E2:59"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="seqfunc.html#E1:59_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="seqfunc.html#T34" target="_self" title="SEQFUNC:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:59"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</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#E3:59"><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span></span><br/>

<i><font color="Green" title="E44">A6</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
 <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">H</font>,<font color="Maroon" title="c4">X</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:59"><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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:59_3"/><i><font color="Green" title="E45">A7</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"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/><a NAME="E2:59_3"/><b>then </b><i><font color="Green" title="E46">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"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/><a NAME="E3:59_3"/><b>then </b><i><font color="Green" title="E47">A9</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#E1:59"><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="E4:59_3"/><b>then </b><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c3">H</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#E1:59"><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/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</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> = 
<font color="Maroon" title="c2">r</font> <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">H</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#E1:59_3"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <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">H</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#E1:59"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="seqfunc.html#E2:59_3"><i><font color="Green" title="E46">A8</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"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</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#E4:59_3"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</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">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</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#E2:59"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="seqfunc.html#E3:59_3"><i><font color="Green" title="E47">A9</font></i></a>, <a class="ref" href="seqfunc.html#T34" target="_self" title="SEQFUNC:th.34">Th34</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:59"/>
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E5:59"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="seqfunc.html#E6:59"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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