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<div class="add">

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a> ;<br/>
<a NAME="E1:4_2_1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_1_1"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a>  )
 ;<br/>

<a NAME="E2:4_2_1_1"/>
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V3">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">E6</font></i>: 
 <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="txt" href="rsspace4.html#E1:4_2_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:4_2_1_1_1"/><i><font color="Green">E7</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="rsspace.html#K2">seq_id</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>2</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T4">RSSPACE:4</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>
<br/>.= 
<span class="p1">(<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#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
;<br/>
<b>hence </b><a NAME="E4:4_2_1_1_1"/>
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rsspace4.html#E2:4_2_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="seqm_3.html#T71">SEQM_3:71</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a> 
 <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:4_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">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">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ;<br/>



<b>assume </b><a NAME="E1:4_2_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a> 
 ;<br/>

<a NAME="E2:4_2_1_2"/>
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V3">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">E6</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T5">RSSPACE:5</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_2_1_2_1"/>
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1_2_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="seqm_3.html#T70">SEQM_3:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K1" target="_self">the_set_of_BoundedRealSequences</a> 
 <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a>, <a class="ref" href="rsspace.html#D7">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" target="_self">the_set_of_BoundedRealSequences</a>  is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:4_2_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>


</div>
