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

<a NAME="E1:41_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font> st ( for <font color="Olive">b<sub>3</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>3</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>5</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>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>1</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>3</sub></font> ) &amp; ( for <font color="Olive">b<sub>3</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>3</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>5</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>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</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>3</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>6</sub></font>, <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><a NAME="E1:41_1_1_1"/><i><font color="Green">E36</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>6</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>and </b><br/><a NAME="E2:41_1_1_1"/><i><font color="Green">E37</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>and </b><br/><a NAME="E3:41_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E4:41_1_1_1"/><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"><font color="Maroon">c<sub>6</sub></font> <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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="normsp_1.html#E3:41_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="normsp_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:41_1_1_1"/><i><font color="Green">E40</font></i>: 
0 <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"><font color="Maroon">c<sub>6</sub></font> <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>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E6:41_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
0 <a href="real_1.html#K6">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4
 
<b>by </b><i><a class="txt" href="normsp_1.html#E4:41_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</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="E8:41_1_1_1"/><i><font color="Green">E42</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" 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>6</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"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4
 <b>by </b><i><a class="txt" href="normsp_1.html#E1:41_1_1_1"><i><font color="Green">E36</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="E10:41_1_1_1"/><i><font color="Green">E43</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>9</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> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4
 <b>by </b><i><a class="txt" href="normsp_1.html#E2:41_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="normsp_1.html#E6:41_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_1_1"/>
<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/><a NAME="E2:41_1_1_1_1"/><b>then </b><i><font color="Green">E45</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>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" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4
 <b>by </b><i><a class="txt" href="normsp_1.html#E8:41_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E3:41_1_1_1_1"/>
<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>9</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> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4
 <b>by </b><i><a class="txt" href="normsp_1.html#E10:41_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E4:41_1_1_1_1"/><b>then </b><i><font color="Green">E46</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>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" 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>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="normsp_1.html#E2:41_1_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:41_1_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K6">-</a> <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>9</sub></font></span>)</span></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>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>
 <b>by </b><i><a class="ref" href="normsp_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E6:41_1_1_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#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>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" 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>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>
 <b>by </b><i><a class="ref" href="normsp_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E7:41_1_1_1_1"/><b>then </b>
not <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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>
 <b>by </b><i><a class="txt" href="normsp_1.html#E4:41_1_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:41_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="normsp_1.html#E4:41_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="normsp_1.html#E5:41_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:41_1_1_1_2"/><b>then </b><i><font color="Green">E45</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>8</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> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4
 <b>by </b><i><a class="txt" href="normsp_1.html#E10:41_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E3:41_1_1_1_2"/>
<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>8</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" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4
 <b>by </b><i><a class="txt" href="normsp_1.html#E8:41_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E4:41_1_1_1_2"/><b>then </b><i><font color="Green">E46</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>8</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" 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>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</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> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="normsp_1.html#E2:41_1_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:41_1_1_1_2"/>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K6">-</a> <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>8</sub></font></span>)</span></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>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</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>
 <b>by </b><i><a class="ref" href="normsp_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E6:41_1_1_1_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#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>8</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" 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>5</sub></font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</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>
 <b>by </b><i><a class="ref" href="normsp_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E7:41_1_1_1_2"/><b>then </b>
not <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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>
 <b>by </b><i><a class="txt" href="normsp_1.html#E4:41_1_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:41_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="normsp_1.html#E4:41_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="normsp_1.html#E5:41_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:41_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="normsp_1.html#E11:41_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font> st ( for <font color="Olive">b<sub>3</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>3</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>5</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>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>1</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>3</sub></font> ) &amp; ( for <font color="Olive">b<sub>3</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>3</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>5</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>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</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>3</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div>
