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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</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="E1:29"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="metric_6.html#R1" target="_self">is_convergent_in_metrspace_to</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:29"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="tbsp_1.html#V2">convergent</a> &amp;  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="metric_6.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="metric_6.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E4:29"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="tbsp_1.html#V2">convergent</a> &amp;  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="metric_6.html#E2:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="metric_6.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="metric_6.html#K3" target="_self">sequence_of_dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="metric_6.html#E3:29"><i><font color="Green">E24</font></i></a>, <a class="ref" href="metric_6.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E6:29"/><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> holds <span class="p1">(<span class="default"><a href="metric_6.html#K3" target="_self">sequence_of_dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<span class="p1">(<span class="default"><a href="metric_6.html#K3" target="_self">sequence_of_dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="metric_6.html#D15" target="_self">Def15</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>
;<br/>
<b>hence </b><a NAME="E2:29_1"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K3" target="_self">sequence_of_dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:29"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="metric_6.html#K3" target="_self">sequence_of_dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E8:29"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><a href="metric_6.html#K3" target="_self">sequence_of_dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E1:29_2"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K3" target="_self">sequence_of_dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="metric_6.html#E6:29"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:29"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="metric_6.html#E5:29"><i><font color="Green">E25</font></i></a>, <a class="txt" href="metric_6.html#E7:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<b>hence </b><a NAME="E10:29"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>


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