<?xml version="1.0"?>
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<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="normsp_1.html#V4" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4" target="_self">convergent</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E4:36"/><i><font color="Green">E33</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#NR3" target="_self">&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" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <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="normsp_1.html#E1:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="normsp_1.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E6:36"/><i><font color="Green">E34</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#NR3" target="_self">&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" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <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="normsp_1.html#E2:36"><i><font color="Green">E32</font></i></a>, <a class="ref" href="normsp_1.html#D9" target="_self">Def9</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="normsp_1.html#D9" target="_self">NORMSP_1:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E7:36"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E8:36"/><b>then </b><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K6">/</a> 2
 
<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>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:36"/><i><font color="Green">E36</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>11</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" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="normsp_1.html#E4:36"><i><font color="Green">E33</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="E12:36"/><i><font color="Green">E37</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>12</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" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="normsp_1.html#E6:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="normsp_1.html#E8:36"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>12</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E13:36"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E14:36"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E15:36"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="normsp_1.html#E13:36"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:36"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="normsp_1.html#E10:36"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E17:36"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E18:36"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="normsp_1.html#E13:36"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:36"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="normsp_1.html#E12:36"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E20:36"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</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>10</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="normsp_1.html#E16:36"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<i><font color="Green">E41</font></i>: <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
;<br/>
<a NAME="E22:36"/>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E23:36"/>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="normsp_1.html#E20:36"><i><font color="Green">E40</font></i></a>, <a class="txt" href="normsp_1.html#E21:36"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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