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<b>let </b><font color="Maroon" title="c1">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) )
 ;<br/>

<a NAME="E2:40"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T40" title="RFUNCT_1:th.40">RFUNCT_1:40</a></i>;<br/>
<a NAME="E3:40"/><i><font color="Green" title="E29">A3</font></i>: 
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="limfunc3.html#T36" target="_self" title="LIMFUNC3:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E4:40"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc3.html#R1" title="LIMFUNC3:pred.1">is_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="limfunc3.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="limfunc3.html#T37" target="_self" title="LIMFUNC3:th.37">Th37</a></i>;<br/>

<b>thus </b> <a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> = 
<span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="limfunc3.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="limfunc3.html#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="limfunc3.html#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="limfunc3.html#T37" target="_self" title="LIMFUNC3:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="limfunc3.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="limfunc3.html#T36" target="_self" title="LIMFUNC3:th.36">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="limfunc3.html#K1" title="LIMFUNC3:func.1">lim</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span>

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