<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> , <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>



<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:17_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:17_1"/>
 <a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font> is <a href="comseq_2.html#V1">bounded</a>
 <b>by </b><i/>;<br/><a NAME="E4:17_1"/><b>then </b><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   <a href="complex1.html#K5">0c</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K1">the_set_of_ComplexSequences</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> 
 <b>by </b><i>, , <a class="ref" href="csspace.html#D6">CSSPACE:def 6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> 
 ;<br/><a NAME="E2:17_2"/><b>then </b><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">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i>, <a class="ref" href="csspace.html#D6">CSSPACE:def 6</a></i>;<br/><b>thus </b><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E3:17"/><i><font color="Green">E54</font></i>: 
 <a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font> is <a href="comseq_2.html#V1">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:17"/><i><font color="Green">E78</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_3"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:17_3"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</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"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> 1</span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<a NAME="E3:17_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 1
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E4:17_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">y</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:17"/><i><font color="Green">E82</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:17"/><i><font color="Green">E83</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="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</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"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><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"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</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"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span>)</span> <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="comseq_1.html#D14">COMSEQ_1:def 14</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"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <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#D4">COMSEQ_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E2:17_4"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</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"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:17"/><i><font color="Green">E84</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="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><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="E1:17_5"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</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="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"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</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>
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>
<a NAME="E2:17_5"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <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"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</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="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"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</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>
 
<b>by </b><i/>;<br/>
<a NAME="E3:17_5"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></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"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></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"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</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>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E4:17_5"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></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"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:17"/><i><font color="Green">E85</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="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><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="E1:17_6"/>
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E2:17_6"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></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"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:17"/>
 <a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font> is <a href="comseq_2.html#V1">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E11:17"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E12:17"/>
 <a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font> is <a href="comseq_2.html#V1">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E13:17"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E14:17"/><i><font color="Green">E88</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_1"><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="E1:17_7_1"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></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"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E2:17_7_1"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/><a NAME="E3:17_7_1"/><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:17_7_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:17_7_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></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"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E6:17_7_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></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"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_7"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">y</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E15:17"/><i><font color="Green">E92</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="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_8"><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"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<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> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span> <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="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<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"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</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>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</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">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E2:17_8"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="comseq_1.html#K4">(#)</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:17"/>
 <a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font> is <a href="comseq_2.html#V1">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E17:17"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E18:17"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T7">PSCOMP_1:7</a></i>;<br/>
<a NAME="E19:17"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E20:17"/><i><font color="Green">E94</font></i>: 
0 <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>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="comseq_1.html#K4">(#)</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="comseq_1.html#K4">(#)</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="seq_1.html#K14">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="integra2.html#K1">*</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#T17">INTEGRA2:17</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">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="integra2.html#T13">INTEGRA2:13</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="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E22:17"/>
( ( <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  implies <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 0 ) &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">y</font></span><a href="clvect_1.html#K5">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</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> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">x</font></span><a href="clvect_1.html#K5">.||</a></span> )
 <b>by </b><i>, , , , , <a class="txt" href="csspace4.html#E1"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>


</div>
