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<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="E1:121"/><i><font color="Green" title="E68">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> &amp; <font color="Maroon" title="c3">f2</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  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="b2">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) )
 ;<br/>

<div><i><font color="Green" title="E69">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><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="E1:121_1"/><i><font color="Green" title="E70">A3</font></i>: 
(  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font> &amp;  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font> )
 ;<br/><b>assume </b><a NAME="E2:121_1"/><i><font color="Green" title="E71">A4</font></i>: 
( <font color="Maroon" title="c4">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> &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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> )
 ;<br/><a NAME="E3:121_1"/><i><font color="Green" title="E72">A5</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">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</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="c2">f1</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</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">f2</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="E4:121_1"/><i><font color="Green" title="E73">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">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5" title="SEQ_1:def.5">SEQ_1:def 5</a></i>;<br/><a NAME="E5:121_1"/><b>then </b>
 <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="c2">f1</font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:121_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="limfunc1.html#E3:121_1"><i><font color="Green" title="E72">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:121_1"/><b>then </b><i><font color="Green" title="E74">A7</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="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:121"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="limfunc1.html#E1:121_1"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="limfunc1.html#E2:121_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></i>;<br/><a NAME="E7:121_1"/>
 <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">f2</font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:121_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="limfunc1.html#E3:121_1"><i><font color="Green" title="E72">A5</font></i></a>, <a class="txt" href="limfunc1.html#E4:121_1"><i><font color="Green" title="E73">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="E8:121_1"/><b>then </b><i><font color="Green" title="E75">A8</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="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:121"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="limfunc1.html#E1:121_1"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="limfunc1.html#E2:121_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></i>;<br/><a NAME="E9:121_1"/><b>then </b>
<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#K11" title="SEQ_1:func.11">(#)</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> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E6:121_1"><i><font color="Green" title="E74">A7</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/><b>hence </b><a NAME="E10:121_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</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="limfunc1.html#E2:121_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="limfunc1.html#E4:121_1"><i><font color="Green" title="E73">A6</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>;<br/><b>thus </b> <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#K8" title="SEQ_1:func.8">(#)</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="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="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E2:121_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="limfunc1.html#E4:121_1"><i><font color="Green" title="E73">A6</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E6:121_1"><i><font color="Green" title="E74">A7</font></i></a>, <a class="txt" href="limfunc1.html#E8:121_1"><i><font color="Green" title="E75">A8</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:121"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:121"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="limfunc1.html#D9" target="_self" title="LIMFUNC1:def.9">Def9</a></i>;<br/>

<b>hence </b><a NAME="E4:121"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</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="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:121"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></i>;<br/>


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