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<div class="add">

<b>set </b><font color="Maroon" title="c1">S</font> =  <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ;<br/>
<b>thus </b><a NAME="E1:21_1_1"/>
 <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
 ;<br/>

<b>thus </b><a NAME="E2:21_1_1"/>
 <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a><br/>

<b>thus </b><a NAME="E1:21_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T6" target="_self" title="RSSPACE:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:21_1_1"/>
 <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a><br/>

<b>thus </b><a NAME="E1:21_1_1_2"/>
<font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="ref" href="rsspace.html#T7" target="_self" title="RSSPACE:th.7">Th7</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:21_1_1"/>
 <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a><br/>

<b>thus </b><a NAME="E1:21_1_1_3"/>
 ex <font color="Olive" title="b1">w</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="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> 
 <b>by </b><i><a class="ref" href="rsspace.html#T8" target="_self" title="RSSPACE:th.8">Th8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:21_1_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>, <font color="Olive" title="b3">w</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>  holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T9" target="_self" title="RSSPACE:th.9">Th9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a><br/>

<b>thus </b><a NAME="E6:21_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">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>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T10" target="_self" title="RSSPACE:th.10">Th10</a></i>;<br/>

<b>thus </b><a NAME="E7:21_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">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>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T11" target="_self" title="RSSPACE:th.11">Th11</a></i>;<br/>

<b>let </b><font color="Maroon" title="c2">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>thus </b><a NAME="E8:21_1_1"/>
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="ref" href="rsspace.html#T12" target="_self" title="RSSPACE:th.12">Th12</a></i>;<br/>


</div>
