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<span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>  &amp; <font color="Maroon" title="c3">f2</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">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="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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) implies ( <font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>  &amp;  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
<font color="Maroon" title="c2">f1</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> 
 <span class="kw">and </span><br/><a NAME="E2:98"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c3">f2</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> 
 <span class="kw">and </span><br/><a NAME="E3:98"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>  &amp;  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E64">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">seq</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</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">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_1"/><span class="lab"><font color="Green" title="E65">A5</font></span>: 
<font color="Maroon" title="c4">seq</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> 
 <span class="kw">and </span><br/><a NAME="E2:98_1"/><span class="lab"><font color="Green" title="E66">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</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">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span> )</span><br/><a NAME="E3:98_1"/><span class="lab"><font color="Green" title="E67">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>;<br/><a NAME="E4:98_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E5:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E3:98_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E6:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A9</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E1:98_1"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></span>;<br/><a NAME="E7:98_1"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font>
 ;<br/><a NAME="E8:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A10</font></span>: 
<font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E1:98_1"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E5:98_1"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></span>;<br/><a NAME="E9:98_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E10:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E3:98_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E11:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A12</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:98_1"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></span>;<br/><a NAME="E12:98_1"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font>
 ;<br/><a NAME="E13:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A13</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:98_1"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E10:98_1"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></span>;<br/><a NAME="E14:98_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98_1"><span class="lab"><font color="Green" title="E70">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:98_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E3:98_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="rfunct_2.html#T8" title="RFUNCT_2:th.8">RFUNCT_2:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span></span><br/><span class="kw">thus </span> <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">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> = 
 <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">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">seq</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E3:98_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="rfunct_2.html#T8" title="RFUNCT_2:th.8">RFUNCT_2:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98_1"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="txt" href="#E11:98_1"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E8:98_1"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="txt" href="#E6:98_1"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="ref" href="seq_2.html#T15" title="SEQ_2:th.15">SEQ_2:15</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:98"/>
<font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="limfunc1.html#D9" target="_self" title="LIMFUNC1:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E6:98"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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