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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E9">A1</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> )
 ;<br/>

<a NAME="E2:13"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="measure6.html#T15" target="_self" title="MEASURE6:th.15">Th15</a></i>;<br/>
<a NAME="E3:13"/>
<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E1:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E2:13"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="measure4.html#T2" title="MEASURE4:th.2">MEASURE4:2</a></i>;<br/>
<a NAME="E5:13"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font>
 ;<br/>

<b>then </b><font color="Maroon" title="c1">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font>
<b>by </b><i><a class="txt" href="measure6.html#E2:13"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="measure6.html#T8" target="_self" title="MEASURE6:th.8">Th8</a></i>
;<br/>
<b>hence </b><a NAME="E3:13_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E1:13"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E4:13"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div>
