<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_2.html#V1" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="clvect_2.html#V1" target="_self">convergent</a>
 ;<br/>

<a NAME="E3:16"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="clvect_2.html#V1" target="_self">convergent</a>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="clvect_2.html#E2:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="clvect_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:16_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:16_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> holds <br/> <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="clvect_2.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="clvect_2.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:16_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> holds <br/> <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="clvect_2.html#E2:16"><i><font color="Green">E16</font></i></a>, <a class="txt" href="clvect_2.html#E2:16_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="clvect_2.html#D2" target="_self">Def2</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E7:16_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E8:16_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:16_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E7:16_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:16_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="clvect_2.html#E4:16_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E11:16_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E12:16_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E7:16_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:16_1"/><b>then </b>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="clvect_2.html#E6:16_1"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E14:16_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="clvect_2.html#E10:16_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E15:16_1"/>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>;<br/><a NAME="E16:16_1"/><b>then </b>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="csspace.html#T60">CSSPACE:60</a></i>;<br/><b>hence </b><a NAME="E17:16_1"/>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E14:16_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
 <a href="clvect_2.html#K1" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="clvect_2.html#D2" target="_self">Def2</a></i>;<br/>


</div>
