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<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:20"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="comseq_2.html#V2" target="_self">convergent</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="comseq_2.html#V2" target="_self">convergent</a> <a href="comseq_1.html#NM1">Complex_Sequence</a> ;<br/>
<a NAME="E3:20"/><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a>
 
;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><a NAME="E1:20_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:20_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:20_1"/><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>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="comseq_2.html#E1:20_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="comseq_2.html#D5" target="_self">Def5</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E5:20_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E6:20_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E4:20_1"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E7:20_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T150">COMPLEX1:150</a></i>;<br/><a NAME="E8:20_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="comseq_2.html#E6:20_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:20_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E10:20_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:20"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="comseq_2.html#E3:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


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