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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:27"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">a1</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b1</font> = <font color="Maroon" title="c3">b</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E5:27"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a> 
 
<b>by </b><i><a class="ref" href="rsspace.html#D1" title="RSSPACE:def.1">RSSPACE:def 1</a></i>;<br/>
<a NAME="E6:27"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c5">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a> 
 
<b>by </b><i><a class="ref" href="rsspace.html#D1" title="RSSPACE:def.1">RSSPACE:def 1</a></i>;<br/>
<a NAME="E7:27"/><i><font color="Green" title="E33">A6</font></i>: 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">a1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E5:27"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="rsspace.html#D2" title="RSSPACE:def.2">RSSPACE:def 2</a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green" title="E34">A7</font></i>: 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c5">b1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E6:27"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="rsspace.html#D2" title="RSSPACE:def.2">RSSPACE:def 2</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="lp_space.html#E2:27"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="lp_space.html#E5:27"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="lp_space.html#E7:27"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="lp_space.html#D2" target="_self" title="LP_SPACE:def.2">Def2</a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green" title="E36">A9</font></i>: 
<font color="Maroon" title="c5">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="lp_space.html#E3:27"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="lp_space.html#E6:27"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="lp_space.html#E8:27"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="lp_space.html#D2" target="_self" title="LP_SPACE:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a1</font> = <font color="Maroon" title="c4">a1</font> as   <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>  <b>by </b><i><a class="txt" href="lp_space.html#E9:27"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c5">b1</font> as   <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>  <b>by </b><i><a class="txt" href="lp_space.html#E10:27"><i><font color="Green" title="E36">A9</font></i></a></i>;<br/>
<a NAME="E13:27"/>
 <a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="lp_space.html#T4" target="_self" title="LP_SPACE:th.4">Th4</a></i>;<br/>
<a NAME="E14:27"/><b>then </b><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E9:27"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="lp_space.html#E10:27"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</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="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">b1</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="c6">a1</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="c7">b1</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/>.= 
<font color="Maroon" title="c2">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="lp_space.html#E7:27"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="lp_space.html#E8:27"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="rsspace.html#T3" title="RSSPACE:th.3">RSSPACE:3</a></i>
;<br/>
<b>hence </b><a NAME="E16:27"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="lp_space.html#E14:27"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="lp_space.html#D2" target="_self" title="LP_SPACE:def.2">Def2</a></i>;<br/>


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