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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</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="E1:4"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="bhsp_2.html#V1" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_2.html#V1" target="_self">convergent</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:4"/><i><font color="Green">E7</font></i>: 
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>2</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#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="bhsp_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:4"/><i><font color="Green">E8</font></i>: 
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>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E2:4"><i><font color="Green">E6</font></i></a>, <a class="ref" href="bhsp_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bhsp_2.html#D1" target="_self">BHSP_2:def 1</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="E7:4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E8:4"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<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="E10:4"/><i><font color="Green">E10</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>2</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#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="bhsp_2.html#E4:4"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:4"/><i><font color="Green">E11</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>9</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#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="bhsp_2.html#E6:4"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bhsp_2.html#E8:4"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E13:4"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E14:4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E15:4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_2.html#E13:4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:4"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="bhsp_1.html#K5">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>11</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_2.html#E10:4"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E17:4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E18:4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_2.html#E13:4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:4"/><b>then </b>
 <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>11</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_2.html#E12:4"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E20:4"/><b>then </b><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="bhsp_2.html#E16:4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E21:4"/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>;<br/>
<a NAME="E22:4"/><b>then </b>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5">dist</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T47">BHSP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E23:4"/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E20:4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
