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<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">f1</font>, <font color="Maroon" title="c3">f2</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="E1:39"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) )
 ;<br/>

<div><i><font color="Green" title="E28">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><a NAME="E1:39_1"/><i><font color="Green" title="E29">A3</font></i>: 
(  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font> )
 ;<br/><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/><b>assume </b><a NAME="E2:39_1"/><i><font color="Green" title="E30">A4</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="relat_1.html#K2" title="RELAT_1:func.2">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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><a NAME="E3:39_1"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><a NAME="E4:39_1"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
( <font color="Maroon" title="c2">f1</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="c2">f1</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="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:39"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="limfunc3.html#E1:39_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="limfunc3.html#E2:39_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="limfunc3.html#D4" target="_self" title="LIMFUNC3:def.4">Def4</a></i>;<br/><a NAME="E5:39_1"/><i><font color="Green" title="E33">A7</font></i>: 
( <font color="Maroon" title="c3">f2</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">f2</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="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:39"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="limfunc3.html#E1:39_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="limfunc3.html#E2:39_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="limfunc3.html#E3:39_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="limfunc3.html#D4" target="_self" title="LIMFUNC3:def.4">Def4</a></i>;<br/><a NAME="E6:39_1"/><i><font color="Green" title="E34">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</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="txt" href="limfunc3.html#E3:39_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>;<br/><b>hence </b><a NAME="E7:39_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</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="limfunc3.html#E4:39_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="limfunc3.html#E5:39_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/><b>thus </b><a NAME="E8:39_1"/>
 <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">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:39_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="limfunc3.html#E5:39_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="limfunc3.html#E6:39_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:39"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="limfunc3.html#D1" target="_self" title="LIMFUNC3:def.1">Def1</a></i>;<br/>

<b>hence </b><a NAME="E4:39"/>
 <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></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="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:39"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="limfunc3.html#D4" target="_self" title="LIMFUNC3:def.4">Def4</a></i>;<br/>


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