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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f1</font> is <a href="limfunc1.html#V4" title="LIMFUNC1:attr.4">divergent_in+infty_to+infty</a> &amp; <font color="Maroon" title="c2">f2</font> is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+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="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">g</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> ) ) implies ( <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#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a> &amp;  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font> ) )</span><br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c1">f1</font> is <a href="limfunc1.html#V4" title="LIMFUNC1:attr.4">divergent_in+infty_to+infty</a> &amp; <font color="Maroon" title="c2">f2</font> is <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+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="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">g</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> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a> &amp;  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font> )</span><br/>

<div><i><font color="Green" title="E6">A2</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" title="c3">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">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> implies ( <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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font> ) )</span><br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font> )</span><br/><a NAME="E2:55_1"/><b>then </b><i><font color="Green" title="E8">A4</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="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="E3:55_1"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</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="#E1:55"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E1:55_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="limfunc1.html#D7" title="LIMFUNC1:def.7">LIMFUNC1:def 7</a></i>;<br/><a NAME="E4:55_1"/>
 <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font>
 ;<br/><a NAME="E5:55_1"/><b>then </b>
( <font color="Maroon" title="c2">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font> )
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:55_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E3:55_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="limfunc1.html#D12" title="LIMFUNC1:def.12">LIMFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E6:55_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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font> )
 <b>by </b><i><a class="txt" href="#E1:55_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="valued_0.html#T31" title="VALUED_0:th.31">VALUED_0:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
<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#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="limfunc1.html#D6" title="LIMFUNC1:def.6">LIMFUNC1:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font></span><br/>

<b>hence </b><a NAME="E4:55"/>
 <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c2">f2</font>
 <b>by </b><i><a class="txt" href="#E2:55"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="limfunc1.html#D12" title="LIMFUNC1:def.12">LIMFUNC1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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