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<b>let </b><font color="Maroon" title="c1">x0</font>, <font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">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="E1:60"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 ;<br/>

<div><i><font color="Green" title="E35">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:60_1"/><i><font color="Green" title="E36">A3</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c1">x0</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="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="c3">f</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:60"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="limfunc2.html#D4" target="_self" title="LIMFUNC2:def.4">Def4</a></i>;<br/><b>take </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font>;<br/><b>thus </b><a NAME="E4:60_1"/>
( <font color="Maroon" title="c6">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c6">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">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:60_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E36">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><a NAME="E1:60_2"/><i><font color="Green" title="E37">A5</font></i>: 
 <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">x0</font>
 ;<br/><b>assume </b><a NAME="E2:60_2"/><i><font color="Green" title="E38">A6</font></i>: 
( <font color="Maroon" title="c4">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="c4">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="c4">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">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</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="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span> )
 ;<br/><a NAME="E3:60_2"/><b>then </b><i><font color="Green" title="E39">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">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="c3">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#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/><a NAME="E4:60_2"/><b>then </b><i><font color="Green" title="E40">A8</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">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="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:60"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="limfunc2.html#E1:60_2"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="limfunc2.html#E2:60_2"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="limfunc2.html#D8" target="_self" title="LIMFUNC2:def.8">Def8</a></i>;<br/><a NAME="E5:60_2"/><b>then </b><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</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/><a NAME="E6:60_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">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#K4" title="LIMFUNC1:func.4">right_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> <font color="Maroon" title="c3">f</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="E7:60_2"/><b>then </b><i><font color="Green" title="E42">A10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">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="c3">f</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:60_2"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:60_2"/><b>then </b><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></i>;<br/><b>thus </b><a NAME="E9:60_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">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#E5:60_2"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="limfunc2.html#E7:60_2"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></i>;<br/><b>thus </b><a NAME="E10:60_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">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E4:60_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="limfunc2.html#E8:60_2"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:60"/>
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:60"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="limfunc2.html#D4" target="_self" title="LIMFUNC2:def.4">Def4</a></i>;<br/>

<b>hence </b><a NAME="E5:60"/>
 <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E3:60"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="limfunc2.html#D8" target="_self" title="LIMFUNC2:def.8">Def8</a></i>;<br/>


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