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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:11"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:11"><i><font color="Green">E14</font></i></a>, <a class="ref" href="limfunc2.html#T1">LIMFUNC2:1</a></i>;<br/>
<a NAME="E5:11"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1:11"><i><font color="Green">E14</font></i></a>, <a class="txt" href="limfunc3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:11"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:11"><i><font color="Green">E14</font></i></a>, <a class="ref" href="limfunc2.html#T2">LIMFUNC2:2</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E8:11"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E9:11"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E10:11"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E8:11"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:11"/>
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:11"><i><font color="Green">E15</font></i></a></i>;<br/>

<a NAME="E12:11"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E13:11"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E8:11"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:11"><i><font color="Green">E16</font></i></a></i>;<br/>


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