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

<b>given </b><font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c4">g2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><a NAME="E2:42_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">g1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 <b>and </b><br/><a NAME="E3:42_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 <b>and </b><br/><a NAME="E4:42_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">g2</font>
 ;<br/>

<div><i><font color="Green" title="E32">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:42_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">g2</font>
 <b>by </b><i><a class="ref" href="toprns_1.html#T25" target="_self" title="TOPRNS_1:th.25">Th25</a></i>;<br/><b>then </b><font color="Maroon" title="c4">g2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">g2</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T51" title="EUCLID:th.51">EUCLID:51</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T43" title="EUCLID:th.43">EUCLID:43</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T32" title="EUCLID:th.32">EUCLID:32</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/><b>hence </b><a NAME="E4:42_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprns_1.html#E4:42_1_1"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:42_1_1"/><i><font color="Green" title="E33">A6</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
;<br/>
<a NAME="E7:42_1_1"/><b>then </b><i><font color="Green" title="E34">A7</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 
<b>by </b><i><a class="txt" href="toprns_1.html#E5:42_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:42_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">g1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:42_1_1"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:42_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 <b>by </b><i><a class="txt" href="toprns_1.html#E3:42_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="toprns_1.html#E7:42_1_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<a NAME="E12:42_1_1"/><i><font color="Green" title="E37">A10</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">g1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 
<b>by </b><i><a class="txt" href="toprns_1.html#E9:42_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E13:42_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
 
<b>by </b><i><a class="txt" href="toprns_1.html#E11:42_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E14:42_1_1"/><b>then </b><i><font color="Green" title="E38">A11</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">g1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E12:42_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T32" title="EUCLID:th.32">EUCLID:32</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p4">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T43" title="EUCLID:th.43">EUCLID:43</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T51" title="EUCLID:th.51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T50" title="EUCLID:th.50">EUCLID:50</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T50" title="EUCLID:th.50">EUCLID:50</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T51" title="EUCLID:th.51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T48" title="EUCLID:th.48">EUCLID:48</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T48" title="EUCLID:th.48">EUCLID:48</a></i>
;<br/>
<a NAME="E16:42_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">g1</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T30" target="_self" title="TOPRNS_1:th.30">Th30</a></i>;<br/>
<a NAME="E17:42_1_1"/><b>then </b><i><font color="Green" title="E39">A12</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">g1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n2</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T13" title="EUCLID:th.13">EUCLID:13</a></i>;<br/>
<a NAME="E18:42_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">g2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E5:42_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="toprns_1.html#E6:42_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<b>hence </b><a NAME="E19:42_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprns_1.html#E14:42_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="toprns_1.html#E17:42_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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