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

<b>set </b><font color="Maroon" title="c1">W</font> =  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> ;<br/>
<a NAME="E1:4_2_1"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  )
 ;<br/>

<a NAME="E2:4_2_1_1"/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">u</font></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:4_2_1_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a></i>;<br/>
<a NAME="E2:4_2_1_1_1"/><i><font color="Green" title="E7">A4</font></i>: 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a></i>;<br/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">u</font></span>)</span> = 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T4" title="RSSPACE:th.4">RSSPACE:4</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3" title="RSSPACE:th.3">RSSPACE:3</a></i>
;<br/>
<b>hence </b><a NAME="E4:4_2_1_1_1"/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">u</font></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_1_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="rsspace4.html#E2:4_2_1_1_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="seqm_3.html#T71" title="SEQM_3:th.71">SEQM_3:71</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_2_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:4_2_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>   st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ;<br/>



<b>assume </b><a NAME="E1:4_2_1_2"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 ;<br/>

<a NAME="E2:4_2_1_2"/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:4_2_1_2_1"/><i><font color="Green" title="E6">A6</font></i>: 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a></i>;<br/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> = 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T5" title="RSSPACE:th.5">RSSPACE:5</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3" title="RSSPACE:th.3">RSSPACE:3</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_2_1_2_1"/>
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_2_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="seqm_3.html#T70" title="SEQM_3:th.70">SEQM_3:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_2_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_2_1"/>
 <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a></i>;<br/>


</div>
