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

<span class="kw">set </span><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"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0: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#K4" title="RSSPACE:func.4">Linear_Space_of_RealSequences</a>  st <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K4" title="RSSPACE:func.4">Linear_Space_of_RealSequences</a>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  holds <br/><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><br/><span class="kw">let </span><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#K4" title="RSSPACE:func.4">Linear_Space_of_RealSequences</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  implies <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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:4_2_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><br/>

<a NAME="E2:4_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace4.html#D1" target="_self" onmouseover="rs('rsspace4/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:4_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">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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T3" onmouseover="rs('rsspace/T3')" onmouseout="rh()">RSSPACE:3</a></span>;<br/>
<a NAME="E4:4_2_1_1"/><span class="kw">then </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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_2_1_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="seqm_3.html#T37" onmouseover="rs('seqm_3/T37')" onmouseout="rh()">SEQM_3:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:4_2_1_1"/>
<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace4.html#D1" target="_self" onmouseover="rs('rsspace4/D1')" onmouseout="rh()">Def1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:4_2_1"/>
 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#K4" title="RSSPACE:func.4">Linear_Space_of_RealSequences</a>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K4" title="RSSPACE:func.4">Linear_Space_of_RealSequences</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  implies <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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:4_2_1_2"/>
( <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><br/>

<a NAME="E2:4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A3</font></span>: 
(  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace4.html#D1" target="_self" onmouseover="rs('rsspace4/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:4_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">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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T2" onmouseover="rs('rsspace/T2')" onmouseout="rh()">RSSPACE:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:4_2_1_2"/>
<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_2_1_2"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="rsspace4.html#D1" target="_self" onmouseover="rs('rsspace4/D1')" onmouseout="rh()">Def1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_2_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="rlsub_1.html#D1" onmouseover="rs('rlsub_1/D1')" onmouseout="rh()">RLSUB_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
