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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:27"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:27"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<b>by </b><i><a class="ref" href="rsspace.html#D1">RSSPACE:def 1</a></i>;<br/>
<a NAME="E6:27"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<b>by </b><i><a class="ref" href="rsspace.html#D1">RSSPACE:def 1</a></i>;<br/>
<a NAME="E7:27"/><i><font color="Green">E38</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rsspace.html#D2">RSSPACE:def 2</a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green">E39</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rsspace.html#D2">RSSPACE:def 2</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="lp_space.html#E2:27"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lp_space.html#E7:27"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="lp_space.html#E3:27"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lp_space.html#E8:27"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E11:27"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lp_space.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="lp_space.html#E9:27"><i><font color="Green">E40</font></i></a>, <a class="ref" href="lp_space.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green">E33</font></i></a>, <a class="txt" href="lp_space.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="lp_space.html#E10:27"><i><font color="Green">E41</font></i></a>, <a class="ref" href="lp_space.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <b>by </b><i><a class="txt" href="lp_space.html#E11:27"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <b>by </b><i><a class="txt" href="lp_space.html#E12:27"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E15:27"/>
 <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lp_space.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E16:27"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lp_space.html#E11:27"><i><font color="Green">E42</font></i></a>, <a class="txt" href="lp_space.html#E12:27"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>
<a NAME="E17:27"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:27"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lp_space.html#D2" target="_self">Def2</a></i>;<br/>
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</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>6</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>7</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/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lp_space.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="lp_space.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
;<br/>
<b>hence </b><a NAME="E19:27"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="txt" href="lp_space.html#E17:27"><i><font color="Green">E44</font></i></a></i>;<br/>


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