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<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">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="E1:10"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c1">f1</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> &amp; <font color="Maroon" title="c2">f2</font> is <a href="limfunc1.html#V5" title="LIMFUNC1:attr.5">divergent_in+infty_to-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">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c1">f1</font></span>)</span> ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c3">s</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="c3">s</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">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c1">f1</font></span>)</span> )
 ;<br/><a NAME="E2:10_1"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s</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="c1">f1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</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">f2</font> )
 <b>by </b><i><a class="txt" href="limfunc4.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/><a NAME="E3:10_1"/><b>then </b>
<font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="limfunc4.html#E1:10_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="limfunc1.html#D10" title="LIMFUNC1:def.10">LIMFUNC1:def 10</a></i>;<br/><a NAME="E4:10_1"/><b>then </b>
<font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font></span>)</span> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="limfunc4.html#E2:10_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="limfunc1.html#D8" title="LIMFUNC1:def.8">LIMFUNC1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:10_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="rfunct_2.html#T39" title="RFUNCT_2:th.39">RFUNCT_2:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
<font color="Maroon" title="c2">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c1">f1</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc4.html#E1:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="limfunc1.html#D11" title="LIMFUNC1:def.11">LIMFUNC1:def 11</a></i>;<br/>


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