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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq</font> <a href="bhsp_3.html#NR1" target="_self">is_Cauchy_sequence</a> 
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="bhsp_3.html#K2" target="_self">^\</a> <font color="Olive">k</font>
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:54"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="bhsp_3.html#K2" target="_self">^\</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="bhsp_3.html#D1" target="_self">BHSP_3:def 1</a><br/>

<b>assume </b><a NAME="E5:54"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">l1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:54"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l1</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l1</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">l</font> = <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E8:54"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">m1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:54"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:54"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">m2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:54"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m2</font> = <font color="Maroon">m2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E16:54"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:54"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">l2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l1</font>
 ;<br/><a NAME="E2:54_1"/><b>then </b>
<font color="Maroon">l2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:54_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">l3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E21:54"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_2"/>
<font color="Maroon">l3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l1</font>
 ;<br/><a NAME="E2:54_2"/><b>then </b>
<font color="Maroon">l3</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:54_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E23:54"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">l2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E24:54"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">l3</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E25:54"/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l3</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E26:54"/><b>then </b>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l3</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E27:54"/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/>


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