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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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="E1:55"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V4">divergent_in+infty_to+infty</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="limfunc1.html#V3">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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</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="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) ) )
 ;<br/>

<div><i><font color="Green">E6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V1">divergent_to+infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</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="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/><a NAME="E2:55_1"/><b>then </b><i><font color="Green">E8</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E3:55_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:55"><i><font color="Green">E5</font></i></a>, <a class="txt" href="limfunc4.html#E1:55_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="limfunc1.html#D7">LIMFUNC1:def 7</a></i>;<br/><a NAME="E4:55_1"/>
 <a href="limfunc1.html#K5">lim_in+infty</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K5">lim_in+infty</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E5:55_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K5">lim_in+infty</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:55"><i><font color="Green">E5</font></i></a>, <a class="txt" href="limfunc4.html#E2:55_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="limfunc4.html#E3:55_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="limfunc1.html#D12">LIMFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E6:55_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K5">lim_in+infty</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:55_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="rfunct_2.html#T39">RFUNCT_2:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="limfunc1.html#V3">convergent_in+infty</a>
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:55"><i><font color="Green">E5</font></i></a>, <a class="ref" href="limfunc1.html#D6">LIMFUNC1:def 6</a></i>;<br/>

<b>hence </b><a NAME="E4:55"/>
 <a href="limfunc1.html#K5">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K5">lim_in+infty</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="limfunc4.html#E2:55"><i><font color="Green">E6</font></i></a>, <a class="ref" href="limfunc1.html#D12">LIMFUNC1:def 12</a></i>;<br/>


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