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<span class="kw">let </span><font color="Maroon" title="c1">p</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b2">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>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Olive" title="b2">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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b2">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>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Olive" title="b2">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> </span><br/>

<span class="kw">let </span><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>; <a class="txt" 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="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>  &amp; <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>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</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>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</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> </span><br/>

<span class="kw">reconsider </span><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="E5:26"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D1" title="RSSPACE:def.1">RSSPACE:def 1</a></span>;<br/>
<a NAME="E6:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D2" title="RSSPACE:def.2">RSSPACE:def 2</a></span>;<br/>
<a NAME="E7:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="lp_space.html#D2" target="_self" title="LP_SPACE:def.2">Def2</a></span>;<br/>
<a NAME="E8:26"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D1" title="RSSPACE:def.1">RSSPACE:def 1</a></span>;<br/>
<a NAME="E9:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D2" title="RSSPACE:def.2">RSSPACE:def 2</a></span>;<br/>
<a NAME="E10:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="lp_space.html#D2" target="_self" title="LP_SPACE:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E32">A6</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E36">A10</font></span>:  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">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="c7">a1</font></span>)</span> <a href="series_1.html#K1" title="SERIES_1:func.1">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T2" title="RSSPACE:th.2">RSSPACE:2</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="series_1.html#K1" title="SERIES_1:func.1">+</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="rsspace.html#T1" title="RSSPACE:th.1">RSSPACE:1</a></span>
;<br/>
<a NAME="E14:26"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lp_space.html#T4" target="_self" title="LP_SPACE:th.4">Th4</a></span>;<br/>
<a NAME="E15:26"/><span class="kw">then </span>
<font color="Maroon" title="c7">a1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:26"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="series_1.html#K1" title="SERIES_1:func.1">+</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="lp_space.html#D2" target="_self" title="LP_SPACE:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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