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<b>let </b><font color="Maroon">c</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="comseq_2.html#V1">bounded</a>
 ;<br/>

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:5"/><i><font color="Green">E33</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i>, <a class="ref" href="comseq_2.html#T8">COMSEQ_2:8</a></i>;<br/>
<a NAME="E4:5"/>
 ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1"><b>case </b></a><a NAME="E1:5_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_1"/><i><font color="Green">E47</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>   holds <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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_1"><b>proof </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/>

<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">c</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:5_1_1_1_1_1"/>
<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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>;<br/>


</div><b>end;</b></div><b>take </b><font color="Maroon">r1</font> = 1;<br/><b>thus </b><a NAME="E3:5_1_1_1_1"/>
 ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2"><b>case </b></a><a NAME="E1:5_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_2"/><i><font color="Green">E53</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>   holds <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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_2_1"><b>proof </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="E1:5_1_1_1_2_1"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> )
 
<b>by </b><i>, , <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
;<br/>
<b>hence </b><a NAME="E3:5_1_1_1_2_1"/>
<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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>


</div><b>end;</b></div><b>take </b><font color="Maroon">r1</font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E3:5_1_1_1_2"/>
 ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5_1"/>
 ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <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</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:5"/>
<font color="Maroon">c</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">seq</font> is <a href="comseq_2.html#V1">bounded</a>
 <b>by </b><i><a class="ref" href="comseq_2.html#D3">COMSEQ_2:def 3</a></i>;<br/>


</div>
