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<b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f1</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>let </b><font color="Maroon">f2</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>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><a NAME="E1:73"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f1</font> <a href="limfunc2.html#R4" target="_self">is_right_convergent_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f2</font> <a href="limfunc2.html#R4" target="_self">is_right_convergent_in</a> <font color="Maroon">x0</font> &amp;  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f2</font>,<font color="Maroon">x0</font> &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Maroon">x0</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> ) ) )
 ;<br/>

<b>given </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:73"/><i><font color="Green">E33</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; ( for <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> ) ) )
 <b>and </b><br/><a NAME="E3:73"/><i><font color="Green">E38</font></i>: 
( ( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> ) or ( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_1"/>
( ( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> ) or ( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i><a class="txt" href="limfunc2.html#E7"><i><font color="Green">Lemma42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_1_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_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="E1:73_1_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</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#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/><a NAME="E2:73_1_1_1_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#T2" target="_self">Th2</a>, </i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:73_1_1_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E5:73_1_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</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"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="limfunc2.html#T3" target="_self">Th3</a>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/><a NAME="E6:73_1_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>;<br/><a NAME="E7:73_1_1_1_1"/>
( <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#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <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#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E8:73_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
(  <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> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="limfunc2.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:73_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:73_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:73_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:73_1_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:73_1_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><a NAME="E4:73_1_1_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:73_1_1_1_1_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E6:73_1_1_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><a NAME="E7:73_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="txt" href="limfunc2.html#E8:73_1_1_1_1"><i><font color="Green">E53</font></i></a>, </i>;<br/><a NAME="E8:73_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/><a NAME="E9:73_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> )
 ;<br/><a NAME="E10:73_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="limfunc2.html#E9:73_1_1_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>hence </b><a NAME="E11:73_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E13:73_1_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E14:73_1_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E15:73_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:73_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:73_1_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E5:73_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:73_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:73_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="limfunc2.html#E9:73_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:73_1_1_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E6:73_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E19:73_1_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="limfunc1.html#T11">LIMFUNC1:11</a></i>;<br/><a NAME="E20:73_1_1_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E8:73_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:73_1_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc2.html#E10:73_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E22:73_1_1_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
( <font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:73_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="limfunc2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E23:73_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="limfunc2.html#T6" target="_self">Th6</a>, <a class="ref" href="limfunc2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E24:73_1_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f2</font>,<font color="Maroon">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:73_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="limfunc2.html#T5" target="_self">Th5</a>, </i>;<br/><div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:73_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:73_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="limfunc2.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E3:73_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E4:73_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="limfunc2.html#E10:73_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="limfunc2.html#T6" target="_self">Th6</a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div><a NAME="E26:73_1_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:73_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="limfunc2.html#D2" target="_self">Def2</a>, <a class="ref" href="limfunc2.html#D3" target="_self">Def3</a>, <a class="ref" href="seq_2.html#T33">SEQ_2:33</a></i>;<br/><a NAME="E27:73_1_1_1_1"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:73_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="limfunc2.html#D2" target="_self">Def2</a>, <a class="ref" href="limfunc2.html#D3" target="_self">Def3</a>, <a class="ref" href="limfunc2.html#D4" target="_self">Def4</a>, <a class="ref" href="seq_2.html#T34">SEQ_2:34</a></i>;<br/><a NAME="E28:73_1_1_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
( <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="seqm_3.html#K1">^\</a> <font color="Maroon">k</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_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D5" target="_self">Def5</a>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>;<br/><b>hence </b><a NAME="E29:73_1_1_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/><b>thus </b><a NAME="E30:73_1_1_1_1"/>
 <a href="seq_2.html#K2">lim</a> <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="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#D6" target="_self">Def6</a>, <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:73_1_1_1"/>
<font color="Maroon">f</font> <a href="limfunc2.html#R4" target="_self">is_right_convergent_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E2:73_1_1_1"><i><font color="Green">E40</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:73_1_1_1"/>
( <font color="Maroon">f</font> <a href="limfunc2.html#R4" target="_self">is_right_convergent_in</a> <font color="Maroon">x0</font> &amp;  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_1_2"/><i><font color="Green">E102</font></i>: 
( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><div><i><font color="Green">E103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_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="E1:73_1_1_2_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</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#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/><a NAME="E2:73_1_1_2_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#T2" target="_self">Th2</a>, </i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:73_1_1_2_1"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E5:73_1_1_2_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</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"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="limfunc2.html#T3" target="_self">Th3</a>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/><a NAME="E6:73_1_1_2_1"/><i><font color="Green">E107</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>;<br/><a NAME="E7:73_1_1_2_1"/>
( <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#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <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#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E8:73_1_1_2_1"/><b>then </b><i><font color="Green">E108</font></i>: 
(  <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> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="limfunc2.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:73_1_1_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:73_1_1_2_1"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:73_1_1_2_1"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:73_1_1_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:73_1_1_2_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><a NAME="E4:73_1_1_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:73_1_1_2_1_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E6:73_1_1_2_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><a NAME="E7:73_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:73_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/><a NAME="E9:73_1_1_2_1_1"/><b>then </b>
 ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> )
 ;<br/><a NAME="E10:73_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E11:73_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E13:73_1_1_2_1"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E14:73_1_1_2_1"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</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#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E15:73_1_1_2_1"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:73_1_1_2_1"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:73_1_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:73_1_1_2_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E19:73_1_1_2_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="limfunc1.html#T11">LIMFUNC1:11</a></i>;<br/><a NAME="E20:73_1_1_2_1"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:73_1_1_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="limfunc2.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E22:73_1_1_2_1"/><b>then </b><i><font color="Green">E121</font></i>: 
( <font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="limfunc2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E23:73_1_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E24:73_1_1_2_1"/><b>then </b><i><font color="Green">E122</font></i>: 
( <font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f2</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="limfunc2.html#T5" target="_self">Th5</a>, </i>;<br/><div><i><font color="Green">E123</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:73_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:73_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="limfunc2.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E3:73_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E4:73_1_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="limfunc2.html#T6" target="_self">Th6</a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div><a NAME="E26:73_1_1_2_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D2" target="_self">Def2</a>, <a class="ref" href="limfunc2.html#D3" target="_self">Def3</a>, <a class="ref" href="seq_2.html#T33">SEQ_2:33</a></i>;<br/><a NAME="E27:73_1_1_2_1"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D2" target="_self">Def2</a>, <a class="ref" href="limfunc2.html#D3" target="_self">Def3</a>, <a class="ref" href="limfunc2.html#D4" target="_self">Def4</a>, <a class="ref" href="seq_2.html#T34">SEQ_2:34</a></i>;<br/><a NAME="E28:73_1_1_2_1"/><b>then </b><i><font color="Green">E125</font></i>: 
( <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="seqm_3.html#K1">^\</a> <font color="Maroon">k</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_2.html#K1">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D5" target="_self">Def5</a>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>;<br/><b>hence </b><a NAME="E29:73_1_1_2_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/><b>thus </b><a NAME="E30:73_1_1_2_1"/>
 <a href="seq_2.html#K2">lim</a> <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="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="limfunc2.html#D6" target="_self">Def6</a>, <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:73_1_1_2"/>
<font color="Maroon">f</font> <a href="limfunc2.html#R4" target="_self">is_right_convergent_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:73_1_1_2"/>
( <font color="Maroon">f</font> <a href="limfunc2.html#R4" target="_self">is_right_convergent_in</a> <font color="Maroon">x0</font> &amp;  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:73"/>
( <font color="Maroon">f</font> <a href="limfunc2.html#R4" target="_self">is_right_convergent_in</a> <font color="Maroon">x0</font> &amp;  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="limfunc2.html#K2" target="_self">lim_right</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> )
 ;<br/>


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