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

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> holds <br/><font color="Olive" title="b1">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> implies <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> )</span><br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> )</span><br/>

<b>assume </b><a NAME="E2:8"/>
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:8"/><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">b</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>case </b></a><a NAME="E1:8_1_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:8_1_1_1_1"/><i><font color="Green" title="E12">A17</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="urysohn2.html#T6" target="_self" title="URYSOHN2:th.6">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c6">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="urysohn2.html#T6" target="_self" title="URYSOHN2:th.6">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">s</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E5:8_1_1_1_1"/><i><font color="Green" title="E13">A18</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_1_1_1"/><i><font color="Green" title="E14">A19</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">q</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:8_1_1_1_1_1_1"/><i><font color="Green" title="E15">A20</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</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="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c11">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c13">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:8_1_1_1_1_1_1"/><i><font color="Green" title="E16">A21</font></i>: 
( <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">z2</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1_1_1_1"><i><font color="Green" title="E15">A20</font></i></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">z2</font> = <font color="Maroon" title="c13">z2</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:8_1_1_1_1_1_1"/><i><font color="Green" title="E17">A22</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">z2</font> &amp; <font color="Maroon" title="c14">z2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="#E4:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E4:8_1_1_1_1_1_1"><i><font color="Green" title="E16">A21</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">1o</font>, <font color="Maroon" title="c16">1r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:8_1_1_1_1_1_1"/><i><font color="Green" title="E18">A23</font></i>: 
( <font color="Maroon" title="c15">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">1r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z2</font> &amp; <font color="Maroon" title="c15">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">1r</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1_1"><i><font color="Green" title="E12">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">2o</font>, <font color="Maroon" title="c18">2r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:8_1_1_1_1_1_1"/><i><font color="Green" title="E19">A24</font></i>: 
( <font color="Maroon" title="c17">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z2</font> &amp; <font color="Maroon" title="c18">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c17">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">2r</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1_1"><i><font color="Green" title="E12">A17</font></i></a>, <a class="txt" href="#E6:8_1_1_1_1_1_1"><i><font color="Green" title="E17">A22</font></i></a></i>;<br/>
<a NAME="E11:8_1_1_1_1_1_1"/><i><font color="Green" title="E20">A25</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">2r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">2o</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E10:8_1_1_1_1_1_1"><i><font color="Green" title="E19">A24</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/>
<a NAME="E12:8_1_1_1_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">1o</font> is   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> &amp; <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">1r</font> is   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">1o1</font>, <font color="Maroon" title="c20">1r1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E14:8_1_1_1_1_1_1"/><i><font color="Green" title="E21">A26</font></i>: 
( <font color="Maroon" title="c19">1o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">1o</font> &amp; <font color="Maroon" title="c20">1r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">1r</font> )
 ;<br/>
<a NAME="E15:8_1_1_1_1_1_1"/>
<font color="Maroon" title="c20">1r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">1o1</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E8:8_1_1_1_1_1_1"><i><font color="Green" title="E18">A23</font></i></a>, <a class="txt" href="#E14:8_1_1_1_1_1_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/>
<b>hence </b><a NAME="E16:8_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E4:8_1_1_1_1_1_1"><i><font color="Green" title="E16">A21</font></i></a>, <a class="txt" href="#E8:8_1_1_1_1_1_1"><i><font color="Green" title="E18">A23</font></i></a>, <a class="txt" href="#E10:8_1_1_1_1_1_1"><i><font color="Green" title="E19">A24</font></i></a>, <a class="txt" href="#E11:8_1_1_1_1_1_1"><i><font color="Green" title="E20">A25</font></i></a>, <a class="txt" href="#E14:8_1_1_1_1_1_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</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>
<a NAME="E2:8_1_1_1_1_1"/>
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">q</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> or <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<b>assume </b><a NAME="E1:8_1_1_1_1_1_2"/><i><font color="Green" title="E15">A27</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c11">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">q1</font> = <font color="Maroon" title="c12">q</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:8_1_1_1_1_1_2"/><i><font color="Green" title="E16">A28</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">q1</font> &amp; <font color="Maroon" title="c13">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c13">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_1_1_1_2"><i><font color="Green" title="E15">A27</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E5:8_1_1_1_1_1_2"/><i><font color="Green" title="E17">A29</font></i>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">q2</font> = <font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E6:8_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c15">q3</font> = <font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E2:8_1_1_1_1_1_2_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E18">A30</font></i>: 
<font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:8_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E19">A31</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q1</font> &amp; <font color="Maroon" title="c16">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">o</font> )
 <b>by </b><i><a class="txt" href="#E4:8_1_1_1_1_1_2"><i><font color="Green" title="E16">A28</font></i></a></i>;<br/>
<a NAME="E3:8_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">o</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E2:8_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E19">A31</font></i></a>, <a class="ref" href="xreal_1.html#T75" title="XREAL_1:th.75">XREAL_1:75</a></i>;<br/>
<b>hence </b><a NAME="E4:8_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E2:8_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E19">A31</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</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>
<a NAME="E2:8_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c13">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">d</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_1_1_1_2"><i><font color="Green" title="E15">A27</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">r</font>, <font color="Maroon" title="c17">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:8_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E19">A32</font></i>: 
( <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q1</font> &amp; <font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c16">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">o</font> )
 ;<br/>
<a NAME="E4:8_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E5:8_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font>
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E3:8_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E19">A32</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</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>
<b>hence </b><a NAME="E3:8_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1_1_1_2_1_1"><i><font color="Green" title="E18">A30</font></i></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>
<b>hence </b><a NAME="E3:8_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</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="#E4:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</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>
<b>hence </b><a NAME="E7:8_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E5:8_1_1_1_1_1_2"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 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>
<b>hence </b><a NAME="E3:8_1_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">d</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1_1_1"><i><font color="Green" title="E14">A19</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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><a NAME="E6:8_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="#E4:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E1:8_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/><b>hence </b><a NAME="E7:8_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 <b>by </b><i><a class="txt" href="#E5:8_1_1_1_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</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><b>hence </b><a NAME="E3:8_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ; <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="8_1_1_2"><b>case </b></a><a NAME="E1:8_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a></span><br/><div class="add"><b>hence </b><a NAME="E2:8_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></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="8_1_1_3"><b>case </b></a><a NAME="E1:8_1_1_3"/><i><font color="Green" title="E11">A44</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_3_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 </span><b>by </b><i><a class="ref" href="urysohn2.html#T6" target="_self" title="URYSOHN2:th.6">Th6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:8_1_1_3_1_1_1"/><i><font color="Green" title="E12">A58</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c5">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c6">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E1:8_1_1_3_1_1_1"><i><font color="Green" title="E12">A58</font></i></a></i>;<br/><a NAME="E4:8_1_1_3_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">s</font> is   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">d</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E6:8_1_1_3_1_1_1"/><i><font color="Green" title="E13">A59</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">s</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E8:8_1_1_3_1_1_1"/><i><font color="Green" title="E14">A60</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_3_1_1_1_1"/><i><font color="Green" title="E15">A61</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">q</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:8_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E16">A62</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</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="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c11">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c13">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:8_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E17">A63</font></i>: 
( <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">z2</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E16">A62</font></i></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">z2</font> = <font color="Maroon" title="c13">z2</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:8_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E18">A64</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">z2</font> &amp; <font color="Maroon" title="c14">z2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="#E4:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E4:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E17">A63</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">1o</font>, <font color="Maroon" title="c16">1r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:8_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E19">A65</font></i>: 
( <font color="Maroon" title="c15">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">1r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z2</font> &amp; <font color="Maroon" title="c15">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">1r</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_3_1_1_1"><i><font color="Green" title="E12">A58</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">2o</font>, <font color="Maroon" title="c18">2r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:8_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E20">A66</font></i>: 
( <font color="Maroon" title="c17">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z2</font> &amp; <font color="Maroon" title="c18">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c17">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">2r</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_3_1_1_1"><i><font color="Green" title="E12">A58</font></i></a>, <a class="txt" href="#E6:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E18">A64</font></i></a></i>;<br/>
<a NAME="E11:8_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E21">A67</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">2r</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_3"><i><font color="Green" title="E11">A44</font></i></a>, <a class="txt" href="#E10:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E20">A66</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E12:8_1_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">1o</font> is   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> &amp; <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">1r</font> is   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">1o1</font>, <font color="Maroon" title="c20">1r1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E14:8_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E22">A68</font></i>: 
( <font color="Maroon" title="c19">1o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">1o</font> &amp; <font color="Maroon" title="c20">1r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">1r</font> )
 ;<br/>
<a NAME="E15:8_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c19">1o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">1r1</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_3"><i><font color="Green" title="E11">A44</font></i></a>, <a class="txt" href="#E8:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E19">A65</font></i></a>, <a class="txt" href="#E14:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E22">A68</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E16:8_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E6:8_1_1_3_1_1_1"><i><font color="Green" title="E13">A59</font></i></a>, <a class="txt" href="#E4:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E17">A63</font></i></a>, <a class="txt" href="#E8:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E19">A65</font></i></a>, <a class="txt" href="#E10:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E20">A66</font></i></a>, <a class="txt" href="#E11:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E21">A67</font></i></a>, <a class="txt" href="#E14:8_1_1_3_1_1_1_1_1"><i><font color="Green" title="E22">A68</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</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>
<a NAME="E2:8_1_1_3_1_1_1_1"/>
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">q</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> or <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<b>assume </b><a NAME="E1:8_1_1_3_1_1_1_1_2"/><i><font color="Green" title="E16">A69</font></i>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c11">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E6:8_1_1_3_1_1_1"><i><font color="Green" title="E13">A59</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:8_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c12">q</font> is   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">q1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:8_1_1_3_1_1_1_1_2"/><i><font color="Green" title="E17">A70</font></i>: 
<font color="Maroon" title="c13">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font>
 ;<br/>
<a NAME="E6:8_1_1_3_1_1_1_1_2"/><i><font color="Green" title="E18">A71</font></i>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">q2</font> = <font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E7:8_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_3_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> is   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">q3</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:8_1_1_3_1_1_1_1_2_1"/><i><font color="Green" title="E19">A72</font></i>: 
<font color="Maroon" title="c15">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 ;<br/>
<a NAME="E4:8_1_1_3_1_1_1_1_2_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_3_1_1_1_1_2_1_1"/><i><font color="Green" title="E20">A73</font></i>: 
<font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1_1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_3_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c13">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">g</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E16">A69</font></i></a>, <a class="txt" href="#E5:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E17">A70</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:8_1_1_3_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E21">A74</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q1</font> &amp; <font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c16">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">o</font> )
 <b>by </b><i><a class="txt" href="#E5:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E17">A70</font></i></a></i>;<br/>
<a NAME="E4:8_1_1_3_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c16">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">o</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_3"><i><font color="Green" title="E11">A44</font></i></a>, <a class="txt" href="#E3:8_1_1_3_1_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A74</font></i></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></i>;<br/>
<b>hence </b><a NAME="E5:8_1_1_3_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E1:8_1_1_3"><i><font color="Green" title="E11">A44</font></i></a>, <a class="txt" href="#E5:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E17">A70</font></i></a>, <a class="txt" href="#E3:8_1_1_3_1_1_1_1_2_1"><i><font color="Green" title="E19">A72</font></i></a>, <a class="txt" href="#E3:8_1_1_3_1_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A74</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</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>
<a NAME="E2:8_1_1_3_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3_1_1_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_3_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">q1</font>
 
<b>by </b><i><a class="txt" href="#E1:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E16">A69</font></i></a>, <a class="txt" href="#E5:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E17">A70</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">ri</font>, <font color="Maroon" title="c17">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:8_1_1_3_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E21">A75</font></i>: 
( <font color="Maroon" title="c16">ri</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q1</font> &amp; <font color="Maroon" title="c16">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">o</font> )
 <b>by </b><i><a class="txt" href="#E6:8_1_1_3_1_1_1"><i><font color="Green" title="E13">A59</font></i></a>, <a class="txt" href="#E5:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E17">A70</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:8_1_1_3_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font>
 <b>by </b><i><a class="txt" href="#E1:8_1_1_3"><i><font color="Green" title="E11">A44</font></i></a>, <a class="txt" href="#E6:8_1_1_3_1_1_1"><i><font color="Green" title="E13">A59</font></i></a>, <a class="txt" href="#E5:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E17">A70</font></i></a>, <a class="txt" href="#E6:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E18">A71</font></i></a>, <a class="txt" href="#E3:8_1_1_3_1_1_1_1_2_1"><i><font color="Green" title="E19">A72</font></i></a>, <a class="txt" href="#E3:8_1_1_3_1_1_1_1_2_1_1_2"><i><font color="Green" title="E21">A75</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</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>
<b>hence </b><a NAME="E3:8_1_1_3_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q3</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c15">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="txt" href="#E3:8_1_1_3_1_1_1_1_2_1"><i><font color="Green" title="E19">A72</font></i></a>, <a class="txt" href="#E1:8_1_1_3_1_1_1_1_2_1_1"><i><font color="Green" title="E20">A73</font></i></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>
<b>hence </b><a NAME="E5:8_1_1_3_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</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="#E4:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E3:8_1_1_3_1_1_1_1_2_1"><i><font color="Green" title="E19">A72</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</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>
<b>hence </b><a NAME="E8:8_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E6:8_1_1_3_1_1_1_1_2"><i><font color="Green" title="E18">A71</font></i></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 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>
<b>hence </b><a NAME="E3:8_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c10">g</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E1:8_1_1_3_1_1_1_1"><i><font color="Green" title="E15">A61</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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><a NAME="E9:8_1_1_3_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="#E4:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E1:8_1_1_3"><i><font color="Green" title="E11">A44</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E10:8_1_1_3_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 <b>by </b><i><a class="txt" href="#E6:8_1_1_3_1_1_1"><i><font color="Green" title="E13">A59</font></i></a>, <a class="txt" href="#E8:8_1_1_3_1_1_1"><i><font color="Green" title="E14">A60</font></i></a>, <a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</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="E3:8_1_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ; <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="E7:8"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
