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<b>let </b><font color="Maroon" title="c2">f</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="E1:87"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> or <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_1"/>
( <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a> or <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a> )
 </span><b>by </b><i><a class="txt" href="limfunc1.html#E1:87"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_1_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a>
 ;<br/><div class="add"><div><i><font color="Green" title="E49">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:87_1_1_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c4">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> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:87_1_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font>;<br/><b>thus </b><a NAME="E3:87_1_1_1_1"/>
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c5">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"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:87_1_1_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:87_1_1_1_2"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c3">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="c3">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"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:87_1_1_1_2"/><b>then </b><i><font color="Green" title="E51">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">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">f</font>
 <b>by </b><i><a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/><a NAME="E3:87_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:87_1_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="limfunc1.html#E1:87_1_1_1_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/><a NAME="E4:87_1_1_1_2"/><b>then </b>
 <a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T52" target="_self" title="LIMFUNC1:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E5:87_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:87_1_1_1_2"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:87_1_1_1"/>
 <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:87_1_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_1_2"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V8" title="LIMFUNC1:attr.8">divergent_in-infty_to-infty</a>
 ;<br/><div class="add"><div><i><font color="Green" title="E49">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:87_1_1_2_1"/><i><font color="Green" title="E50">A9</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c4">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> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:87_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/><b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font>;<br/><b>thus </b><a NAME="E3:87_1_1_2_1"/>
( <font color="Maroon" title="c5">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c5">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"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:87_1_1_2_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:87_1_1_2_2"/><i><font color="Green" title="E50">A10</font></i>: 
( <font color="Maroon" title="c3">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="c3">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"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ;<br/><a NAME="E2:87_1_1_2_2"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">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">f</font>
 <b>by </b><i><a class="ref" href="seq_1.html#D10" title="SEQ_1:def.10">SEQ_1:def 10</a></i>;<br/><a NAME="E3:87_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:87_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="limfunc1.html#E1:87_1_1_2_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="limfunc1.html#D11" target="_self" title="LIMFUNC1:def.11">Def11</a></i>;<br/><a NAME="E4:87_1_1_2_2"/><b>then </b>
 <a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T52" target="_self" title="LIMFUNC1:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E5:87_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:87_1_1_2_2"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="rfunct_2.html#T25" title="RFUNCT_2:th.25">RFUNCT_2:25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:87_1_1_2"/>
 <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2:87_1_1_2"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="limfunc1.html#D10" target="_self" title="LIMFUNC1:def.10">Def10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:87"/>
 <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c2">f</font> is <a href="limfunc1.html#V7" title="LIMFUNC1:attr.7">divergent_in-infty_to+infty</a>
 ;<br/>


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