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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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="E1:120"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="limfunc1.html#V6" target="_self">convergent_in-infty</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V6" target="_self">convergent_in-infty</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a>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>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<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="seq_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) ) )
 ;<br/>

<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:120_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</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="seq_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:120"><i><font color="Green">E65</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E3:120_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</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="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:120_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="rfunct_1.html#T40">RFUNCT_1:40</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:120"/><i><font color="Green">E67</font></i>: 
(  <a href="seq_1.html#K16">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V6" target="_self">convergent_in-infty</a> &amp;  <a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:120"><i><font color="Green">E65</font></i></a>, <a class="ref" href="limfunc1.html#T125" target="_self">Th125</a></i>;<br/>
<a NAME="E4:120"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="limfunc1.html#V6" target="_self">convergent_in-infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:120"><i><font color="Green">E65</font></i></a>, <a class="txt" href="limfunc1.html#E2:120"><i><font color="Green">E66</font></i></a>, <a class="ref" href="limfunc1.html#T126" target="_self">Th126</a></i>;<br/>
<b>hence </b><a NAME="E5:120"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V6" target="_self">convergent_in-infty</a>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T40">RFUNCT_1:40</a></i>;<br/>

<b>thus </b> <a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T40">RFUNCT_1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="limfunc1.html#E1:120"><i><font color="Green">E65</font></i></a>, <a class="txt" href="limfunc1.html#E2:120"><i><font color="Green">E66</font></i></a>, <a class="txt" href="limfunc1.html#E3:120"><i><font color="Green">E67</font></i></a>, <a class="ref" href="limfunc1.html#T126" target="_self">Th126</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" target="_self">lim_in-infty</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

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