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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>





<b>assume </b><a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/>

<a NAME="E2:2"/><i><font color="Green">E4</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="measure4.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:2_1_1_1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:2_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E3:2_1_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:2_1_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E3:2_1_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="measure4.html#E6:2_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E3:2_1_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="measure4.html#E6:2_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E3:2_1_1_1_1"><i><font color="Green">E6</font></i></a></i>
;<br/><b>hence </b><a NAME="E8:2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_2"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="measure4.html#E1:2_1_1_1_2"><i><font color="Green">E6</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_2_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> = 
<a href="supinf_1.html#K2">-infty</a>  <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E1:2_1_1_1_2_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_2_1_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> = 
<a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E1:2_1_1_1_2_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E1:2_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E5:2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="measure4.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:2_2_1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:2_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E3:2_2_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:2_2_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E3:2_2_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="measure4.html#E6:2_2_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E3:2_2_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="measure4.html#E6:2_2_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E3:2_2_1_1"><i><font color="Green">E6</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font> = 
<a href="supinf_1.html#K2">-infty</a>  <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E1:2_2_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_3"><b>suppose </b></a><a NAME="E1:2_2_1_3"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font> = 
<a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E1:2_2_1_3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="measure4.html#E4:2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/>


</div>
