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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="rsspace4.html#K3" target="_self">linfty_Space</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E13</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_1"/><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:13_1"/>
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:13_1"/><b>then </b><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> holds 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:13_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rsspace4.html#E2:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="rsspace4.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E5:13_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E6:13_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace4.html#K3" target="_self">linfty_Space</a> 
 <b>by </b><i><a class="txt" href="rsspace4.html#E4:13_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rsspace4.html#T3" target="_self">Th3</a>, <a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace4.html#K3" target="_self">linfty_Space</a> 
 ;<br/><a NAME="E2:13_2"/><b>then </b><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a>, <a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>;<br/><b>thus </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="rsspace4.html#E2:13_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="rsspace4.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E3:13"/><i><font color="Green">E14</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:13"/><i><font color="Green">E15</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<a NAME="E1:13_3"/><i><font color="Green">E16</font></i>: 
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E3:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seqm_3.html#T70">SEQM_3:70</a></i>;<br/>
<a NAME="E2:13_3"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<a NAME="E3:13_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</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="E4:13_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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:13_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rsspace4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:13"/><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K1">.||</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:13"/><i><font color="Green">E17</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:13"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_4"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:13"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:13_5"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>
<a NAME="E2:13_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E7:13"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E3:13_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<b>hence </b><a NAME="E4:13_5"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:13"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:13_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E2:13_6"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rsspace4.html#E8:13"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:13"/>
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T70">SEQM_3:70</a></i>;<br/>
<a NAME="E12:13"/>
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E13:13"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T70">SEQM_3:70</a></i>;<br/>
<a NAME="E14:13"/><i><font color="Green">E23</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:13_7_1"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rsspace4.html#E9:13"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E2:13_7_1"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/><a NAME="E3:13_7_1"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E11:13"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rsspace4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E4:13_7_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E13:13"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rsspace4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E5:13_7_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E2:13_7_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rsspace4.html#E3:13_7_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E6:13_7_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:13_7_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:13"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_8"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:13"/>
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E17:13"/><b>then </b>
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T70">SEQM_3:70</a></i>;<br/>
<a NAME="E18:13"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</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="E19:13"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</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="E20:13"/><i><font color="Green">E26</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</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"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</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"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE: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">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace4.html#E15:13"><i><font color="Green">E24</font></i></a>, <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">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra2.html#K1">*</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#T17">INTEGRA2:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</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">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace4.html#E19:13"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rsspace4.html#E20:13"><i><font color="Green">E26</font></i></a>, <a class="ref" href="integra2.html#T13">INTEGRA2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>
;<br/>
<b>hence </b><a NAME="E22:13"/>
( ( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace4.html#K3" target="_self">linfty_Space</a>  ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace4.html#K3" target="_self">linfty_Space</a>  implies <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 ) &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> )
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rsspace4.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rsspace4.html#E4:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rsspace4.html#E5:13"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rsspace4.html#E6:13"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rsspace4.html#E14:13"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rsspace4.html#T3" target="_self">Th3</a></i>;<br/>


</div>
