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<div class="add">

<span class="kw">let </span><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>; <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">f</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>  implies (  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:96"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</font></span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E60">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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="c3">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="c3">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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:96_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c3">seq</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> 
 <span class="kw">and </span><br/><a NAME="E2:96_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</font></span>)</span> )</span><br/><a NAME="E3:96_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/><a NAME="E4:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A6</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">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">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">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></span>;<br/><a NAME="E5:96_1"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c2">f</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">f</font>
 ;<br/><a NAME="E6:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">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="#E1:96"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E3:96_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="limfunc1.html#D13" target="_self" title="LIMFUNC1:def.13">Def13</a></span>;<br/><a NAME="E7:96_1"/><span class="kw">then </span>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">seq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ;<br/><span class="kw">hence </span><a NAME="E8:96_1"/>
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">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="#E3:96_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">seq</font></span>)</span> = 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E4:96_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="seq_4.html#T14" title="SEQ_4:th.14">SEQ_4:14</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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="Olive" title="b1">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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c4">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:96_2"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
( <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="limfunc1.html#D9" target="_self" title="LIMFUNC1:def.9">Def9</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font>; <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="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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E3:96_2"/>
( <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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_2"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</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="E4:96"/>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</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="#E2:96"><span class="lab"><font color="Green" title="E60">A2</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"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E5:96"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E60">A2</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/>


</div>
