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

<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E19">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>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D4" title="XXREAL_0:def.4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="measure6.html#E2:26"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="supinf_2.html#T19" title="SUPINF_2:th.19">SUPINF_2:19</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><a NAME="E1:26_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:26_1_1"/>
not <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c2">eps</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E4:26_1_1"/><i><font color="Green" title="E22">A4</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="c2">eps</font> &amp; <font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c2">eps</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp; <font color="Maroon" title="c2">eps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="measure6.html#T24" target="_self" title="MEASURE6:th.24">Th24</a></i>;<br/><b>consider </b><font color="Maroon" title="c3">eps0</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E6:26_1_1"/><i><font color="Green" title="E23">A5</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="c3">eps0</font> &amp; <font color="Maroon" title="c3">eps0</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">eps</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E4:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="measure6.html#T26" target="_self" title="MEASURE6:th.26">Th26</a></i>;<br/><a NAME="E7:26_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:26_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><a NAME="E8:26_1_1"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T1" title="SUPINF_1:th.1">SUPINF_1:1</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:26_1_1"/><i><font color="Green" title="E25">A7</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="c3">eps0</font> &amp; <font color="Maroon" title="c3">eps0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">eps</font> &amp; <font color="Maroon" title="c2">eps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E4:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="measure6.html#E6:26_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="measure6.html#T22" target="_self" title="MEASURE6:th.22">Th22</a></i>;<br/><a NAME="E10:26_1_1"/><b>then </b><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c3">eps0</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#T18" target="_self" title="MEASURE6:th.18">Th18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">a</font> =  <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c3">eps0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="measure6.html#E8:26_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="measure6.html#E9:26_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="measure6.html#T18" target="_self" title="MEASURE6:th.18">Th18</a>, <a class="ref" href="measure6.html#T29" target="_self" title="MEASURE6:th.29">Th29</a></i>;<br/><a NAME="E12:26_1_1"/><i><font color="Green" title="E27">A9</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E13:26_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E6:26_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><a NAME="E14:26_1_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E15:26_1_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E16:26_1_1"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E12:26_1_1"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/><a NAME="E17:26_1_1"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="measure6.html#E6:26_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><a NAME="E18:26_1_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K2" title="MEASURE6:func.2">Seg</a> <font color="Maroon" title="c1">x</font> &amp; not <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1_1_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K2" title="MEASURE6:func.2">Seg</a> <font color="Maroon" title="c1">x</font> holds <br/><font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font>
 ;<br/>

<a NAME="E2:26_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="measure6.html#K2" title="MEASURE6:func.2">Seg</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E3:26_1_1_1"/><b>then </b>
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="measure6.html#K2" title="MEASURE6:func.2">Seg</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a></i>;<br/>
<a NAME="E4:26_1_1_1"/><b>then </b>
 <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font>
 
;<br/>
<b>hence </b><a NAME="E5:26_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E12:26_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="measure6.html#E16:26_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="measure6.html#E17:26_1_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c6">y</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E20:26_1_1"/><i><font color="Green" title="E30">A12</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K2" title="MEASURE6:func.2">Seg</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 ;<br/><a NAME="E21:26_1_1"/><i><font color="Green" title="E31">A13</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="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="measure6.html#E20:26_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="measure6.html#D2" target="_self" title="MEASURE6:def.2">Def2</a></i>;<br/><a NAME="E22:26_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">eps0</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font>
 <b>by </b><i><a class="txt" href="measure6.html#E10:26_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="measure6.html#E20:26_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/><a NAME="E23:26_1_1"/><b>then </b><i><font color="Green" title="E32">A14</font></i>: 
 <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font>
 <b>by </b><i><a class="txt" href="measure6.html#E10:26_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="measure6.html#T8" target="_self" title="MEASURE6:th.8">Th8</a></i>;<br/><a NAME="E24:26_1_1"/>
 <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>  <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="measure6.html#T27" target="_self" title="MEASURE6:th.27">Th27</a></i>;<br/><a NAME="E25:26_1_1"/><b>then </b><i><font color="Green" title="E33">A15</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E23:26_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="measure1.html#T4" title="MEASURE1:th.4">MEASURE1:4</a></i>;<br/><a NAME="E26:26_1_1"/><i><font color="Green" title="E34">A16</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="c6">y</font> &amp;  <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="c3">eps0</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="measure6.html#E6:26_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="measure6.html#E20:26_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="measure6.html#D2" target="_self" title="MEASURE6:def.2">Def2</a></i>;<br/><a NAME="E27:26_1_1"/><b>then </b><i><font color="Green" title="E35">A17</font></i>: 
<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> <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="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font>
 <b>by </b><i><a class="ref" href="measure1.html#T4" title="MEASURE1:th.4">MEASURE1:4</a></i>;<br/><a NAME="E28:26_1_1"/><i><font color="Green" title="E36">A18</font></i>: 
<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> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/><a NAME="E29:26_1_1"/><b>then </b><i><font color="Green" title="E37">A19</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="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="measure6.html#E26:26_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="measure1.html#T4" title="MEASURE1:th.4">MEASURE1:4</a></i>;<br/><a NAME="E30:26_1_1"/><i><font color="Green" title="E38">A20</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="c3">eps0</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font>
 <b>by </b><i><a class="txt" href="measure6.html#E6:26_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="measure6.html#E28:26_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="measure1.html#T4" title="MEASURE1:th.4">MEASURE1:4</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span> = 
<font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">eps0</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure6.html#E26:26_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="measure6.html#E27:26_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="measure6.html#E28:26_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="measure4.html#T1" title="MEASURE4:th.1">MEASURE4:1</a></i>
<br/>.= 
<font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">eps0</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure6.html#E26:26_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="measure4.html#T1" title="MEASURE4:th.1">MEASURE4:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">eps0</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span>
<b>by </b><i><a class="txt" href="measure6.html#E26:26_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="measure6.html#E30:26_1_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="measure4.html#T1" title="MEASURE4:th.1">MEASURE4:1</a></i>
;<br/><a NAME="E32:26_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c3">eps0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c2">eps</font>
 <b>by </b><i><a class="txt" href="measure6.html#E6:26_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="measure6.html#E21:26_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="measure6.html#E29:26_1_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="measure6.html#E30:26_1_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="measure1.html#T4" title="MEASURE1:th.4">MEASURE1:4</a></i>;<br/><b>hence </b><a NAME="E33:26_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E4:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="measure6.html#E25:26_1_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><a NAME="E1:26_1_2"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:26_1_2"/><i><font color="Green" title="E22">A22</font></i>: 
 <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="measure6.html#T28" target="_self" title="MEASURE6:th.28">Th28</a></i>;<br/><a NAME="E3:26_1_2"/>
 <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>  <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="measure6.html#T27" target="_self" title="MEASURE6:th.27">Th27</a></i>;<br/><a NAME="E4:26_1_2"/><b>then </b>
(  <a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:26_1_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/><a NAME="E5:26_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> <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="txt" href="measure6.html#E2:26_1_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="supinf_2.html#T14" title="SUPINF_2:th.14">SUPINF_2:14</a></i>;<br/><b>hence </b><a NAME="E6:26_1_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K3" title="MEASURE6:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:26_1_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="supinf_2.html#D2" title="SUPINF_2:def.2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
