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<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="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_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="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#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>  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="hidden.html#R2" title="HIDDEN: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#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="hidden.html#R2" title="HIDDEN: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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a></span>;<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>
<span class="kw">by</span> <span class="lab"><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></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T3" title="RSSPACE:th.3">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="seq_2.html#V3" title="SEQ_2:attr.3">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#T70" title="SEQM_3:th.70">SEQM_3:70</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="hidden.html#R2" title="HIDDEN: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" title="RSSPACE4:def.1">Def1</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#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_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#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  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="hidden.html#R2" title="HIDDEN: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="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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="hidden.html#R2" title="HIDDEN: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="seq_2.html#V3" title="SEQ_2:attr.3">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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a></span>;<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>
<span class="kw">by</span> <span class="lab"><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></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T3" title="RSSPACE:th.3">RSSPACE:3</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="hidden.html#R2" title="HIDDEN: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" title="RSSPACE4:def.1">Def1</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" title="RLSUB_1:def.1">RLSUB_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
