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

<b>let </b><font color="Maroon" title="c1">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R1" title="LIMFUNC2:pred.1">is_left_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x0</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b2">g</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> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 ;<br/>

<a NAME="E2:57"/><i><font color="Green" title="E33">A2</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E34">A3</font></i>: <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="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c4">g</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> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font>;<br/><a NAME="E4:57_1"/>
not <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:57_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:57_1"/><b>then </b>
not <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:57_1"/>
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c5">g</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">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:57"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="limfunc2.html#E3:57_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E35">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:57_2"/><i><font color="Green" title="E36">A6</font></i>: 
( <font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span> )
 ;<br/><a NAME="E2:57_2"/><i><font color="Green" title="E37">A7</font></i>: 
( <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"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> &amp; <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"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E3:57_2"/><b>then </b><i><font color="Green" title="E38">A8</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:57_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:57_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:57"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E5:57_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:57_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:57_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:57_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><a NAME="E7:57_2"/><b>then </b><i><font color="Green" title="E39">A9</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="limfunc2.html#E1:57_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="limfunc2.html#D7" target="_self" title="LIMFUNC2:def.7">Def7</a></i>;<br/><a NAME="E8:57_2"/><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:57_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="limfunc2.html#E2:57_2"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="rfunct_2.html#T26" title="RFUNCT_2:th.26">RFUNCT_2:26</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:57_2"/><i><font color="Green" title="E41">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:57_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="limfunc2.html#E2:57_2"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="rfunct_2.html#T27" title="RFUNCT_2:th.27">RFUNCT_2:27</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E10:57_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="limfunc2.html#E7:57_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="limfunc2.html#E8:57_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="seq_2.html#T35" title="SEQ_2:th.35">SEQ_2:35</a></i>;<br/><b>thus </b><a NAME="E11:57_2"/>
 <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">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="limfunc2.html#E7:57_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="limfunc2.html#E8:57_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="limfunc2.html#E9:57_2"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="seq_2.html#T36" title="SEQ_2:th.36">SEQ_2:36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a>  <a href="limfunc2.html#R1" title="LIMFUNC2:pred.1">is_left_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:57"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="limfunc2.html#D1" target="_self" title="LIMFUNC2:def.1">Def1</a></i>;<br/>

<b>hence </b><a NAME="E6:57"/>
 <a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="limfunc2.html#E4:57"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="limfunc2.html#D7" target="_self" title="LIMFUNC2:def.7">Def7</a></i>;<br/>


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