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<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="E1:114"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_2.html#V2" target="_self">Cauchy</a>
 <b>and </b><br/><a NAME="E2:114"/><i><font color="Green">E58</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="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="clvect_2.html#K8" 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="E4:114"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="clvect_2.html#K8" target="_self">^\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E2:114"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_2.html#D8" target="_self">CLVECT_2:def 8</a><br/>

<b>assume </b><a NAME="E5:114"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:114"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</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"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E1:114"><i><font color="Green">E57</font></i></a>, <a class="ref" href="clvect_2.html#D8" target="_self">Def8</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</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>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E8:114"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:114"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E11:114"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E10:114"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>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="E13:114"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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="txt" href="clvect_2.html#E8:114"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E14:114"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E13:114"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>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="E16:114"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E11:114"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:114_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:114_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>6</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="E3:114_1"/>
contradiction
 <b>by </b><i><a class="txt" href="clvect_2.html#E8:114"><i><font color="Green">E61</font></i></a>, <a class="txt" href="clvect_2.html#E16:114"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:114"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E14:114"><i><font color="Green">E65</font></i></a></i>;<br/>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:114_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:114_2"/><b>then </b>
<font color="Maroon">c<sub>13</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>6</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="E3:114_2"/>
contradiction
 <b>by </b><i><a class="txt" href="clvect_2.html#E8:114"><i><font color="Green">E61</font></i></a>, <a class="txt" href="clvect_2.html#E19:114"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:114"/><i><font color="Green">E70</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E16:114"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E22:114"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E19:114"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E23:114"/>
 <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>12</sub></font></span>)</span>,<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>13</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E7:114"><i><font color="Green">E60</font></i></a>, <a class="txt" href="clvect_2.html#E17:114"><i><font color="Green">E67</font></i></a>, <a class="txt" href="clvect_2.html#E20:114"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E24:114"/><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>8</sub></font></span>)</span>,<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>13</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E4:114"><i><font color="Green">E59</font></i></a>, <a class="txt" href="clvect_2.html#E21:114"><i><font color="Green">E70</font></i></a>, <a class="ref" href="clvect_2.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E25:114"/>
 <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>8</sub></font></span>)</span>,<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>9</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="clvect_2.html#E4:114"><i><font color="Green">E59</font></i></a>, <a class="txt" href="clvect_2.html#E22:114"><i><font color="Green">E71</font></i></a>, <a class="ref" href="clvect_2.html#D11" target="_self">Def11</a></i>;<br/>


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