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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="E1:79"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="limfunc1.html#V8" target="_self">divergent_in-infty_to-infty</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V8" target="_self">divergent_in-infty_to-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> <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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) ) )
 ;<br/>

<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_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="E2:79_1"/><i><font color="Green">E49</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> <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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79"><i><font color="Green">E47</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="E3:79_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> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:79_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:79_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/><a NAME="E2:79_2"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</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> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/><a NAME="E3:79_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79"><i><font color="Green">E47</font></i></a>, <a class="txt" href="limfunc1.html#E1:79_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E4:79_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79"><i><font color="Green">E47</font></i></a>, <a class="txt" href="limfunc1.html#E1:79_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="limfunc1.html#E2:79_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E5:79_2"/><b>then </b>
<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>4</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:79_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="limfunc1.html#T38" target="_self">Th38</a></i>;<br/><b>hence </b><a NAME="E6:79_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="limfunc1.html#E2:79_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rfunct_2.html#T23">RFUNCT_2:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:79"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V8" target="_self">divergent_in-infty_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:79"><i><font color="Green">E48</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self">Def11</a></i>;<br/>

<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><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="E2:79_3"/><i><font color="Green">E50</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> <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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79"><i><font color="Green">E47</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="E3:79_3"/>
( <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#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:79_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:79_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/><a NAME="E2:79_4"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <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#K8">(#)</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="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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</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> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/><a NAME="E3:79_4"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79"><i><font color="Green">E47</font></i></a>, <a class="txt" href="limfunc1.html#E1:79_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E4:79_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79"><i><font color="Green">E47</font></i></a>, <a class="txt" href="limfunc1.html#E1:79_4"><i><font color="Green">E50</font></i></a>, <a class="txt" href="limfunc1.html#E2:79_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E5:79_4"/><b>then </b>
<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>4</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:79_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="limfunc1.html#T51" target="_self">Th51</a></i>;<br/><b>hence </b><a NAME="E6:79_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:79_4"><i><font color="Green">E50</font></i></a>, <a class="txt" href="limfunc1.html#E2:79_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfunct_2.html#T23">RFUNCT_2:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:79"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V7" target="_self">divergent_in-infty_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E5:79"><i><font color="Green">E49</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self">Def10</a></i>;<br/>


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