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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</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="E1:24"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_1.html#V6">convergent</a>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E23</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K3" target="_self">^\</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:24"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="clopban3.html#E2:24"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:24"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="clopban3.html#E1:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a><br/>

<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="E7:24"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<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="E9:24"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></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="clopban3.html#E6:24"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E10:24"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:24"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E13:24"/>
<font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="txt" href="clopban3.html#E12:24"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:24"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:24_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:24_1"/>
contradiction
 <b>by </b><i><a class="txt" href="clopban3.html#E10:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="clopban3.html#E15:24"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E15:24"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>

;<br/>
<a NAME="E18:24"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></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="clopban3.html#E9:24"><i><font color="Green">E26</font></i></a>, <a class="txt" href="clopban3.html#E16:24"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:24"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></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="clopban3.html#E4:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="clopban3.html#E17:24"><i><font color="Green">E31</font></i></a>, <a class="ref" href="clopban3.html#D6" target="_self">Def6</a></i>;<br/>


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