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

<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">q1</font>, <font color="Maroon" title="c4">q2</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>







<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E5">A1</font></i>: 
<span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ;<br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">q1</font>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T22" target="_self" title="RCOMP_2:th.22">Th22</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c4">q2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">p1</font>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E1:25"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="rcomp_2.html#T22" target="_self" title="RCOMP_2:th.22">Th22</a></i>;<br/>
<b>thus </b><a NAME="E4:25"/>
<span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="rcomp_2.html#E1:25_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<a NAME="E4:25_1"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p2</font> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E2:25_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="rcomp_2.html#T3" target="_self" title="RCOMP_2:th.3">Th3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1"/>
( <font color="Maroon" title="c3">q1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">x2</font> or <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">q2</font> )
 </span><b>by </b><i><a class="txt" href="rcomp_2.html#E2:25_1"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><b>case </b></a><a NAME="E1:25_1_1_1_1"/>
<font color="Maroon" title="c3">q1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">x2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_2.html#E4:25_1"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><b>case </b></a><a NAME="E1:25_1_1_1_2"/>
<font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">q2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_1_2"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_2.html#E4:25_1"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:25_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:25"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 </span><b>by </b><i><a class="txt" href="rcomp_2.html#E5:25"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1"><b>case </b></a><a NAME="E1:25_2_1_1"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E3:25_2_1_1"/><i><font color="Green" title="E10">A9</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q1</font> )
 <b>by </b><i><a class="txt" href="rcomp_2.html#E1:25_2_1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="rcomp_2.html#T3" target="_self" title="RCOMP_2:th.3">Th3</a></i>;<br/><a NAME="E4:25_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p2</font>
 <b>by </b><i><a class="txt" href="rcomp_2.html#E2:25"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:25_2_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_2.html#E3:25_2_1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="rcomp_2.html#T3" target="_self" title="RCOMP_2:th.3">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2"><b>case </b></a><a NAME="E1:25_2_1_2"/><i><font color="Green" title="E9">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E3:25_2_1_2"/><i><font color="Green" title="E10">A11</font></i>: 
( <font color="Maroon" title="c4">q2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="rcomp_2.html#E1:25_2_1_2"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="rcomp_2.html#T3" target="_self" title="RCOMP_2:th.3">Th3</a></i>;<br/><a NAME="E4:25_2_1_2"/><b>then </b>
( <font color="Maroon" title="c1">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="rcomp_2.html#E3:25"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:25_2_1_2"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_2.html#E3:25_2_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="rcomp_2.html#T3" target="_self" title="RCOMP_2:th.3">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:25"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div>
