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

<b>given </b><font color="Maroon">c<sub>3</sub></font>, <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="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><a NAME="E1:47_1_1"/><i><font color="Green">E35</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="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5" 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="E2:47_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="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" 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="E3:47_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:47_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="toprns_1.html#T25" target="_self">Th25</a></i>;<br/><b>then </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T43">EUCLID:43</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T32">EUCLID:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><b>hence </b><a NAME="E4:47_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprns_1.html#E3:47_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:47_1_1"/><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E6:47_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i><a class="txt" href="toprns_1.html#E4:47_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:47_1_1"/><i><font color="Green">E41</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>5</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="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 4
 <b>by </b><i><a class="txt" href="toprns_1.html#E1:47_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:47_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>6</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="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 4
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:47_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="toprns_1.html#E6:47_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E11:47_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E12:47_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i><a class="txt" href="toprns_1.html#E8:47_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E13:47_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E14:47_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i><a class="txt" href="toprns_1.html#E10:47_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E15:47_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5" 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="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" 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="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K6">/</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E12:47_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T32">EUCLID:32</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T43">EUCLID:43</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" 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>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" 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>3</sub></font> <a href="euclid.html#K20">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K20">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T48">EUCLID:48</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T48">EUCLID:48</a></i>
;<br/>
<a NAME="E17:47_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E18:47_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E19:47_1_1"/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" 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="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E4:47_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="toprns_1.html#E5:47_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<b>hence </b><a NAME="E20:47_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprns_1.html#E15:47_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="toprns_1.html#E18:47_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
