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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="limfunc2.html#R1" target="_self">is_left_convergent_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<b>then </b><i><font color="Green">E32</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>

<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>
;<br/>
<a NAME="E3:55"/><b>then </b><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="limfunc2.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="ref" href="limfunc2.html#D1" target="_self">Def1</a></i>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:55_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><a NAME="E2:55_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="limfunc2.html#E2:55"><i><font color="Green">E32</font></i></a>, <a class="ref" href="limfunc2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:55_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E4:55_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:55"><i><font color="Green">E32</font></i></a>, <a class="txt" href="limfunc2.html#E1:55_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:55_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:55"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rfunct_2.html#T26">RFUNCT_2:26</a></i>;<br/><a NAME="E6:55_2"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K18">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:55"><i><font color="Green">E32</font></i></a>, <a class="txt" href="limfunc2.html#E4:55_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rfunct_2.html#T27">RFUNCT_2:27</a></i>;<br/><b>hence </b><a NAME="E7:55_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="limfunc2.html#E2:55_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="limfunc2.html#E5:55_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_2.html#T35">SEQ_2:35</a></i>;<br/><b>thus </b><a NAME="E8:55_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="limfunc2.html#E2:55_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="limfunc2.html#E5:55_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="limfunc2.html#E6:55_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seq_2.html#T36">SEQ_2:36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:55"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a>  <a href="limfunc2.html#R1" target="_self">is_left_convergent_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:55"><i><font color="Green">E33</font></i></a>, <a class="ref" href="limfunc2.html#D1" target="_self">Def1</a></i>;<br/>

<b>hence </b><a NAME="E6:55"/>
 <a href="limfunc2.html#K1" target="_self">lim_left</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K1" target="_self">lim_left</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="limfunc2.html#E4:55"><i><font color="Green">E34</font></i></a>, <a class="ref" href="limfunc2.html#D7" target="_self">Def7</a></i>;<br/>


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