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<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> &amp; <font color="Maroon" title="c1">A</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a> &amp; <font color="Maroon" title="c2">B</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a> &amp;  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:142"/><i><font color="Green" title="E79">Z1</font></i>: 
( <font color="Maroon" title="c1">A</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> &amp; <font color="Maroon" title="c1">A</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a> )
 <b>and </b><br/><a NAME="E2:142"/><i><font color="Green" title="E80">Z2</font></i>: 
<font color="Maroon" title="c2">B</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a>
 <b>and </b><br/><a NAME="E3:142"/><i><font color="Green" title="E81">Z3</font></i>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a></span><br/>

<b>set </b><font color="Maroon" title="c3">z</font> =  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font>;<br/>
<a NAME="E4:142"/><i><font color="Green" title="E82">G1</font></i>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E79">Z1</font></i></a>, <a class="ref" href="xxreal_2.html#D6" target="_self" title="XXREAL_2:def.6">Def6</a>, <a class="txt" href="#E3:142"><i><font color="Green" title="E81">Z3</font></i></a></i>;<br/>
<a NAME="E5:142"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">r</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">r</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:142_1"/><i><font color="Green" title="E83">Z3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:142_1"/><i><font color="Green" title="E84">Z4</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E3:142_1"/><i><font color="Green" title="E85">Z5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 <b>and </b><br/><a NAME="E4:142_1"/><i><font color="Green" title="E86">Z6</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:142_1_1"/>
( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:142_1"><i><font color="Green" title="E83">Z3</font></i></a>, <a class="txt" href="#E2:142_1"><i><font color="Green" title="E84">Z4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_1"><b>suppose </b></a><a NAME="E1:142_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><a NAME="E2:142_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E79">Z1</font></i></a>, <a class="txt" href="#E3:142_1"><i><font color="Green" title="E85">Z5</font></i></a>, <a class="txt" href="#E4:142_1"><i><font color="Green" title="E86">Z6</font></i></a>, <a class="ref" href="xxreal_2.html#T84" target="_self" title="XXREAL_2:th.84">Th84</a></i>;<br/><b>hence </b><a NAME="E3:142_1_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:142_1_1_2"/><i><font color="Green" title="E87">S1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E2:142_1_1_2"/><i><font color="Green" title="E88">S2</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:142_1_1_2_1"/>
( <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> or  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:142_1_1_2_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><a NAME="E2:142_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E79">Z1</font></i></a>, <a class="txt" href="#E1:142_1_1_2"><i><font color="Green" title="E87">S1</font></i></a>, <a class="txt" href="#E4:142"><i><font color="Green" title="E82">G1</font></i></a>, <a class="txt" href="#E3:142_1"><i><font color="Green" title="E85">Z5</font></i></a>, <a class="ref" href="xxreal_2.html#T84" target="_self" title="XXREAL_2:th.84">Th84</a></i>;<br/><b>hence </b><a NAME="E3:142_1_1_2_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:142_1_1_2_1_2"/>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><a NAME="E2:142_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E80">Z2</font></i></a>, <a class="txt" href="#E2:142_1_1_2"><i><font color="Green" title="E88">S2</font></i></a>, <a class="txt" href="#E4:142_1"><i><font color="Green" title="E86">Z6</font></i></a>, <a class="ref" href="xxreal_2.html#T86" target="_self" title="XXREAL_2:th.86">Th86</a></i>;<br/><b>hence </b><a NAME="E3:142_1_1_2_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:142_1_1_3"/><i><font color="Green" title="E87">S1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:142_1_1_3"/><i><font color="Green" title="E88">S2</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:142_1_1_3_1"/>
(  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> or <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:142_1_1_3_1_1"/>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><a NAME="E2:142_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E1:142"><i><font color="Green" title="E79">Z1</font></i></a>, <a class="txt" href="#E2:142_1_1_3"><i><font color="Green" title="E88">S2</font></i></a>, <a class="txt" href="#E4:142"><i><font color="Green" title="E82">G1</font></i></a>, <a class="txt" href="#E4:142_1"><i><font color="Green" title="E86">Z6</font></i></a>, <a class="ref" href="xxreal_2.html#T84" target="_self" title="XXREAL_2:th.84">Th84</a></i>;<br/><b>hence </b><a NAME="E3:142_1_1_3_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:142_1_1_3_1_2"/><i><font color="Green" title="E89">S</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><a NAME="E2:142_1_1_3_1_2"/>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:142_1_1_3"><i><font color="Green" title="E87">S1</font></i></a>, <a class="ref" href="xxreal_2.html#T3" target="_self" title="XXREAL_2:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E3:142_1_1_3_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E1:142_1_1_3_1_2"><i><font color="Green" title="E89">S</font></i></a>, <a class="txt" href="#E3:142_1"><i><font color="Green" title="E85">Z5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_4"><b>suppose </b></a><a NAME="E1:142_1_1_4"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><a NAME="E2:142_1_1_4"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E2:142"><i><font color="Green" title="E80">Z2</font></i></a>, <a class="txt" href="#E3:142_1"><i><font color="Green" title="E85">Z5</font></i></a>, <a class="txt" href="#E4:142_1"><i><font color="Green" title="E86">Z6</font></i></a>, <a class="ref" href="xxreal_2.html#T84" target="_self" title="XXREAL_2:th.84">Th84</a></i>;<br/><b>hence </b><a NAME="E3:142_1_1_4"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:142"/>
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a>
 <b>by </b><i><a class="ref" href="xxreal_2.html#T88" target="_self" title="XXREAL_2:th.88">Th88</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
