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<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:134"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> is <a href="limfunc1.html#V4" target="_self">divergent_in+infty_to+infty</a> or <font color="Maroon">f</font> is <a href="limfunc1.html#V5" target="_self">divergent_in+infty_to-infty</a> )
 <b>and </b><br/><a NAME="E2:134"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E3:134"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:134_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="limfunc1.html#T38" target="_self">Th38</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g</font>;<br/><a NAME="E3:134_1"/>
not <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="limfunc1.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:134_1"/><b>then </b>
not <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:134_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="limfunc1.html#T39" target="_self">Th39</a>, <a class="ref" href="limfunc1.html#T41" target="_self">Th41</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_2_1"/>
( <font color="Maroon">f</font> is <a href="limfunc1.html#V4" target="_self">divergent_in+infty_to+infty</a> or <font color="Maroon">f</font> is <a href="limfunc1.html#V5" target="_self">divergent_in+infty_to-infty</a> )
 </span><b>by </b><i><a class="ref" href="limfunc1.html#T37" target="_self">Th37</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_1_1"><b>suppose </b></a><a NAME="E1:134_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> is <a href="limfunc1.html#V4" target="_self">divergent_in+infty_to+infty</a>
 ;<br/><div class="add"><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:134_2_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 ;<br/><a NAME="E2:134_2_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="limfunc1.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E3:134_2_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E58"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:134_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T41" target="_self">Th41</a>, <a class="txt" href="limfunc1.html#E58"><i><font color="Green">Lemma75</font></i></a>, <a class="txt" href="limfunc1.html#E3:134"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E5:134_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:134"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E6:134_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="limfunc1.html#E58"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="rfunct_2.html#T27">RFUNCT_2:27</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:134_2_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a>  is <a href="limfunc1.html#V3" target="_self">convergent_in+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T40" target="_self">Th40</a>, </i>;<br/><b>hence </b><a NAME="E4:134_2_1_1"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a>  is <a href="limfunc1.html#V3" target="_self">convergent_in+infty</a> &amp;  <a href="limfunc1.html#K5" target="_self">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="limfunc1.html#T47" target="_self">Th47</a>, <a class="ref" href="limfunc1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_1_2"><b>suppose </b></a><a NAME="E1:134_2_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> is <a href="limfunc1.html#V5" target="_self">divergent_in+infty_to-infty</a>
 ;<br/><div class="add"><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:134_2_1_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">seq</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 ;<br/><a NAME="E2:134_2_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="limfunc1.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E3:134_2_1_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="limfunc1.html#E58"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:134_2_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T41" target="_self">Th41</a>, <a class="txt" href="limfunc1.html#E58"><i><font color="Green">Lemma75</font></i></a>, <a class="txt" href="limfunc1.html#E4:134"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E5:134_2_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:134"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E6:134_2_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="limfunc1.html#E58"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="rfunct_2.html#T27">RFUNCT_2:27</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:134_2_1_2"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a>  is <a href="limfunc1.html#V3" target="_self">convergent_in+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T40" target="_self">Th40</a>, </i>;<br/><b>hence </b><a NAME="E4:134_2_1_2"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a>  is <a href="limfunc1.html#V3" target="_self">convergent_in+infty</a> &amp;  <a href="limfunc1.html#K5" target="_self">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="limfunc1.html#T47" target="_self">Th47</a>, <a class="ref" href="limfunc1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:134"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a>  is <a href="limfunc1.html#V3" target="_self">convergent_in+infty</a> &amp;  <a href="limfunc1.html#K5" target="_self">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


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