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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<i><font color="Green">E26</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:32_2"/><i><font color="Green">E28</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="normsp_1.html#K1">||.</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>4</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="real_ns1.html#E2:32"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E5:32_2"/>
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/> <a href="bhsp_1.html#K5">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>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:32_2_1"/><i><font color="Green">E29</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>7</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="normsp_1.html#K1">||.</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>4</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E3:32_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E4:32_2_1"/><i><font color="Green">E30</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>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_2_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <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>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:32_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:32_2_1_1"/><b>then </b><i><font color="Green">E32</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>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
<span class="p1"><a href="normsp_1.html#K1">||.</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>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="real_ns1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="bhsp_1.html#K3">||.</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>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="txt" href="real_ns1.html#E4:32_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E6:32_2_1_1"/>
 <a href="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E3:32_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="real_ns1.html#E1:32_2_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E5:32_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E4:32_2_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:32_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_2.html#V1">convergent</a>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D1">BHSP_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="real_ns1.html#E2:32"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E8:32_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 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>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="bhsp_1.html#K5">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>3</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </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="E1:32_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<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="E3:32_2_2"/><i><font color="Green">E30</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="normsp_1.html#K1">||.</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> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</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="real_ns1.html#E1:32_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>
<a NAME="E4:32_2_2"/><i><font color="Green">E31</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/> <a href="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_2_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <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>9</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:32_2_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:32_2_2_1"/><b>then </b><i><font color="Green">E33</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>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
<span class="p1"><a href="normsp_1.html#K1">||.</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>9</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="real_ns1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="bhsp_1.html#K3">||.</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>9</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="txt" href="real_ns1.html#E4:32_2_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E6:32_2_2_1"/>
 <a href="bhsp_1.html#K5">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>9</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:32_2_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="real_ns1.html#E3:32_2_2"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E5:32_2_2"/>
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="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E4:32_2_2"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:32_2"/>
 <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E6:32_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="bhsp_2.html#D2">BHSP_2:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="normsp_1.html#V4">convergent</a> implies ( <font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_2.html#V1">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_2.html#V1">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:32_3"/><i><font color="Green">E28</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/> <a href="bhsp_1.html#K5">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>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D1">BHSP_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="real_ns1.html#E2:32"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E5:32_3"/>
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="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:32_3_1"/><i><font color="Green">E29</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>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E3:32_3"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E4:32_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>7</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="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:32_3_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E3:32_3_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <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> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:32_3_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:32_3_1_1"/><b>then </b><i><font color="Green">E31</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>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
 <a href="bhsp_1.html#K5">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>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
;<br/>
<b>hence </b><a NAME="E7:32_3_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E2:32_3_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="real_ns1.html#E5:32_3_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="real_ns1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>7</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="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:32_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="real_ns1.html#E2:32"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E8:32_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 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>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</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>6</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_2"><b>proof </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="E1:32_3_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<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="E3:32_3_2"/><i><font color="Green">E30</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="bhsp_1.html#K5">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="bhsp_2.html#K1">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>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1:32_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="bhsp_2.html#D2">BHSP_2:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E4:32_3_2"/>
<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>
 ;<br/>

<a NAME="E5:32_3_2"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="bhsp_1.html#K5">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>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1">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>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E3:32_3_2"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <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="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E7:32_3_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E8:32_3_2"/><b>then </b><i><font color="Green">E32</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>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#T13" target="_self">Th13</a></i>;<br/>
 <a href="bhsp_1.html#K5">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>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_ns1.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
;<br/>
<b>hence </b><a NAME="E10:32_3_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</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>9</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</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="real_ns1.html#E5:32_3_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="real_ns1.html#E8:32_3_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="real_ns1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:32_3"/>
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E6:32_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_2.html#V1">convergent</a> implies ( <font color="Maroon">c<sub>2</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>


</div>
