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<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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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#V3" title="MEASURE5:attr.3">right_open_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#V4" title="MEASURE5:attr.4">left_open_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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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#V3" title="MEASURE5:attr.3">right_open_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#V4" title="MEASURE5:attr.4">left_open_interval</a> )</span><br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E11">A1</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"> (  not <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_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#V4" title="MEASURE5:attr.4">left_open_interval</a> )</span><br/>

<b>assume </b><a NAME="E2:10"/>
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_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#V4" title="MEASURE5:attr.4">left_open_interval</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">a</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> , <font color="Maroon" title="c4">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E4:10"/><i><font color="Green" title="E12">A2</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#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D7" title="MEASURE5:def.7">MEASURE5:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</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="10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
( ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <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="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <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="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <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="10_1_1_1"><b>case </b></a><a NAME="E1:10_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</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#V4" title="MEASURE5:attr.4">left_open_interval</a></span><br/><div class="add"><b>hence </b><a NAME="E2:10_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#V4" title="MEASURE5:attr.4">left_open_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="10_1_1_2"><b>case </b></a><a NAME="E1:10_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">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#V4" title="MEASURE5:attr.4">left_open_interval</a></span><br/><div class="add"><b>hence </b><a NAME="E2:10_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#V4" title="MEASURE5:attr.4">left_open_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="10_1_1_3"><b>case </b></a><a NAME="E1:10_1_1_3"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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#V4" title="MEASURE5:attr.4">left_open_interval</a></span><br/><div class="add"><b>hence </b><a NAME="E2:10_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#V4" title="MEASURE5:attr.4">left_open_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="10_1_1_4"><b>case </b></a><a NAME="E1:10_1_1_4"/><i><font color="Green" title="E13">A20</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="c4">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#V4" title="MEASURE5:attr.4">left_open_interval</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_4"/><i><font color="Green" title="E14">A21</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ;<br/><b>consider </b><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:10_1_1_4"/><i><font color="Green" title="E15">A22</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="#E1:10_1_1_4"><i><font color="Green" title="E13">A20</font></i></a></i>;<br/><a NAME="E6:10_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">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="c8">d</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E8:10_1_1_4"/><i><font color="Green" title="E16">A23</font></i>: 
<font color="Maroon" title="c8">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="c6">s</font>
 ;<br/><a NAME="E9:10_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r</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">g</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E11:10_1_1_4"/><i><font color="Green" title="E17">A24</font></i>: 
<font color="Maroon" title="c9">g</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">r</font>
 ;<br/><a NAME="E12:10_1_1_4"/><i><font color="Green" title="E18">A25</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#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_4_1"/><i><font color="Green" title="E19">A26</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#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="c10">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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:10_1_1_4_1_1"/><i><font color="Green" title="E20">A27</font></i>: 
<font color="Maroon" title="c10">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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c12">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_4_1_1"/><i><font color="Green" title="E21">A28</font></i>: 
( <font color="Maroon" title="c12">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="c11">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="c12">z2</font> )
 <b>by </b><i><a class="txt" href="#E1:10_1_1_4_1_1"><i><font color="Green" title="E20">A27</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="c13">z2</font> = <font color="Maroon" title="c12">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="E6:10_1_1_4_1_1"/><i><font color="Green" title="E22">A29</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="c13">z2</font> &amp; <font color="Maroon" title="c13">z2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E4:10_1_1_4_1_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/>
<a NAME="E7:10_1_1_4_1_1"/><i><font color="Green" title="E23">A30</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="c13">z2</font> &amp; <font color="Maroon" title="c13">z2</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="c13">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E4:10_1_1_4_1_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">1o</font>, <font color="Maroon" title="c15">1r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:10_1_1_4_1_1"/><i><font color="Green" title="E24">A31</font></i>: 
( <font color="Maroon" title="c14">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c15">1r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z2</font> &amp; <font color="Maroon" title="c14">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">1r</font> )
 <b>by </b><i><a class="txt" href="#E1:10_1_1_4"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="#E6:10_1_1_4_1_1"><i><font color="Green" title="E22">A29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">2o</font>, <font color="Maroon" title="c17">2r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E11:10_1_1_4_1_1"/><i><font color="Green" title="E25">A32</font></i>: 
( <font color="Maroon" title="c16">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z2</font> &amp; <font color="Maroon" title="c17">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c16">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">2r</font> )
 <b>by </b><i><a class="txt" href="#E1:10_1_1_4"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="#E7:10_1_1_4_1_1"><i><font color="Green" title="E23">A30</font></i></a></i>;<br/>
<a NAME="E12:10_1_1_4_1_1"/><i><font color="Green" title="E26">A33</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="c15">1r</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="c14">1o</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E9:10_1_1_4_1_1"><i><font color="Green" title="E24">A31</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/>
<a NAME="E13:10_1_1_4_1_1"/><i><font color="Green" title="E27">A34</font></i>: 
<font color="Maroon" title="c16">2o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">2r</font>
 
<b>by </b><i><a class="txt" href="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E4:10_1_1_4_1_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="#E11:10_1_1_4_1_1"><i><font color="Green" title="E25">A32</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/>
<a NAME="E14:10_1_1_4_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="c16">2o</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="c17">2r</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="c18">2o1</font>, <font color="Maroon" title="c19">2r1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E16:10_1_1_4_1_1"/><i><font color="Green" title="E28">A35</font></i>: 
( <font color="Maroon" title="c18">2o1</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">2o</font> &amp; <font color="Maroon" title="c19">2r1</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="c17">2r</font> )
 ;<br/>
<a NAME="E17:10_1_1_4_1_1"/>
<font color="Maroon" title="c19">2r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">2o1</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E13:10_1_1_4_1_1"><i><font color="Green" title="E27">A34</font></i></a>, <a class="txt" href="#E16:10_1_1_4_1_1"><i><font color="Green" title="E28">A35</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E18:10_1_1_4_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="#E3:10_1_1_4"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="#E5:10_1_1_4"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="#E8:10_1_1_4"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="#E11:10_1_1_4"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="#E4:10_1_1_4_1_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="#E9:10_1_1_4_1_1"><i><font color="Green" title="E24">A31</font></i></a>, <a class="txt" href="#E11:10_1_1_4_1_1"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="#E12:10_1_1_4_1_1"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="#E16:10_1_1_4_1_1"><i><font color="Green" title="E28">A35</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</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>
<a NAME="E2:10_1_1_4_1"/>
<span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</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="10_1_1_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> or <font color="Maroon" title="c10">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="E1:10_1_1_4_1_2"/><i><font color="Green" title="E20">A36</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</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="c10">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="c11">q</font> = <font color="Maroon" title="c10">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E8:10_1_1_4"><i><font color="Green" title="E16">A23</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="E3:10_1_1_4_1_2"/>
<font color="Maroon" title="c11">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="c12">q1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_4_1_2"/><i><font color="Green" title="E21">A37</font></i>: 
<font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font>
 ;<br/>
<a NAME="E6:10_1_1_4_1_2"/><i><font color="Green" title="E22">A38</font></i>: 
( <font color="Maroon" title="c9">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font> &amp; <font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c12">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="#E1:10_1_1_4_1_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="#E5:10_1_1_4_1_2"><i><font color="Green" title="E21">A37</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/>
<a NAME="E7:10_1_1_4_1_2"/><i><font color="Green" title="E23">A39</font></i>: 
<font color="Maroon" title="c11">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="c11">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="#E1:10"><i><font color="Green" title="E11">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="c13">q2</font> = <font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E8:10_1_1_4_1_2"/>
<font color="Maroon" title="c11">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="10_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_4_1_2_1"/>
<font color="Maroon" title="c11">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="c14">q3</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:10_1_1_4_1_2_1"/><i><font color="Green" title="E24">A40</font></i>: 
<font color="Maroon" title="c14">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 ;<br/>
<a NAME="E4:10_1_1_4_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="c14">q3</font> &amp; <font color="Maroon" title="c14">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c14">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="10_1_1_4_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_4_1_2_1_1"/><i><font color="Green" title="E25">A41</font></i>: 
<font color="Maroon" title="c14">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_4_1_2_1_1_1"/><i><font color="Green" title="E26">A42</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font>
 
<b>by </b><i><a class="txt" href="#E1:10_1_1_4_1_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="#E5:10_1_1_4_1_2"><i><font color="Green" title="E21">A37</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">p</font>, <font color="Maroon" title="c16">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_4_1_2_1_1_1"/><i><font color="Green" title="E27">A43</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> &amp; <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">o</font> )
 <b>by </b><i><a class="txt" href="#E11:10_1_1_4"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="#E6:10_1_1_4_1_2"><i><font color="Green" title="E22">A38</font></i></a></i>;<br/>
<a NAME="E4:10_1_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c16">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">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="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E1:10_1_1_4_1_2_1_1_1"><i><font color="Green" title="E26">A42</font></i></a>, <a class="txt" href="#E3:10_1_1_4_1_2_1_1_1"><i><font color="Green" title="E27">A43</font></i></a>, <a class="ref" href="xreal_1.html#T77" title="XREAL_1:th.77">XREAL_1:77</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c14">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E5:10_1_1_4"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="#E11:10_1_1_4"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="#E5:10_1_1_4_1_2"><i><font color="Green" title="E21">A37</font></i></a>, <a class="txt" href="#E3:10_1_1_4_1_2_1"><i><font color="Green" title="E24">A40</font></i></a>, <a class="txt" href="#E3:10_1_1_4_1_2_1_1_1"><i><font color="Green" title="E27">A43</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="E2:10_1_1_4_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="c14">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_4_1_2_1_1_2"/>
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">d</font>
 
<b>by </b><i><a class="txt" href="#E1:10_1_1_4_1_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="#E5:10_1_1_4_1_2"><i><font color="Green" title="E21">A37</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">r</font>, <font color="Maroon" title="c16">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_4_1_2_1_1_2"/><i><font color="Green" title="E26">A44</font></i>: 
( <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> &amp; <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">o</font> )
 <b>by </b><i><a class="txt" href="#E8:10_1_1_4"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="#E5:10_1_1_4_1_2"><i><font color="Green" title="E21">A37</font></i></a></i>;<br/>
<a NAME="E4:10_1_1_4_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="c11">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="c11">q</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">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="E5:10_1_1_4_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="c14">q3</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E3:10_1_1_4"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="#E8:10_1_1_4"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="#E5:10_1_1_4_1_2"><i><font color="Green" title="E21">A37</font></i></a>, <a class="txt" href="#E3:10_1_1_4_1_2_1"><i><font color="Green" title="E24">A40</font></i></a>, <a class="txt" href="#E3:10_1_1_4_1_2_1_1_2"><i><font color="Green" title="E26">A44</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="E3:10_1_1_4_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="c14">q3</font> &amp; <font color="Maroon" title="c14">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c14">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="#E3:10_1_1_4_1_2_1"><i><font color="Green" title="E24">A40</font></i></a>, <a class="txt" href="#E1:10_1_1_4_1_2_1_1"><i><font color="Green" title="E25">A41</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="E5:10_1_1_4_1_2_1"/>
<font color="Maroon" title="c11">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="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E3:10_1_1_4_1_2_1"><i><font color="Green" title="E24">A40</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</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>
<b>hence </b><a NAME="E9:10_1_1_4_1_2"/>
<font color="Maroon" title="c10">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="#E7:10_1_1_4_1_2"><i><font color="Green" title="E23">A39</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="E3:10_1_1_4_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#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="#E1:10_1_1_4_1"><i><font color="Green" title="E19">A26</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="E13:10_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">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="c6">s</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E3:10_1_1_4"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="#E5:10_1_1_4"><i><font color="Green" title="E15">A22</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="E14:10_1_1_4"/>
<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#V4" title="MEASURE5:attr.4">left_open_interval</a>
 <b>by </b><i><a class="txt" href="#E8:10_1_1_4"><i><font color="Green" title="E16">A23</font></i></a>, <a class="txt" href="#E11:10_1_1_4"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="#E12:10_1_1_4"><i><font color="Green" title="E18">A25</font></i></a>, <a class="ref" href="measure5.html#D8" title="MEASURE5:def.8">MEASURE5:def 8</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="10_1_1_5"><b>case </b></a><a NAME="E1:10_1_1_5"/><i><font color="Green" title="E13">A45</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="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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#V4" title="MEASURE5:attr.4">left_open_interval</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_5"/><i><font color="Green" title="E14">A46</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ;<br/><b>consider </b><font color="Maroon" title="c7">c</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:10_1_1_5"/><i><font color="Green" title="E15">A47</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ;<br/><a NAME="E6:10_1_1_5"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">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="c8">d</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E8:10_1_1_5"/><i><font color="Green" title="E16">A48</font></i>: 
<font color="Maroon" title="c8">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="c6">s</font>
 ;<br/><a NAME="E9:10_1_1_5"/><i><font color="Green" title="E17">A49</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#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_5_1"/><i><font color="Green" title="E18">A50</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#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="c9">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="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:10_1_1_5_1_1"/><i><font color="Green" title="E19">A51</font></i>: 
<font color="Maroon" title="c9">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="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c11">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:10_1_1_5_1_1"/><i><font color="Green" title="E20">A52</font></i>: 
( <font color="Maroon" title="c11">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="c10">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="c11">z2</font> )
 <b>by </b><i><a class="txt" href="#E1:10_1_1_5_1_1"><i><font color="Green" title="E19">A51</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="c12">z2</font> = <font color="Maroon" title="c11">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/>
<b>reconsider </b><font color="Maroon" title="c13">q</font> = <font color="Maroon" title="c10">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="E7:10_1_1_5_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="c12">z2</font> &amp; <font color="Maroon" title="c12">z2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E4:10_1_1_5_1_1"><i><font color="Green" title="E20">A52</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">o</font>, <font color="Maroon" title="c15">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:10_1_1_5_1_1"/><i><font color="Green" title="E21">A53</font></i>: 
( <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; <font color="Maroon" title="c14">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">r</font> )
 <b>by </b><i><a class="txt" href="#E1:10_1_1_5"><i><font color="Green" title="E13">A45</font></i></a></i>;<br/>
<a NAME="E10:10_1_1_5_1_1"/><i><font color="Green" title="E22">A54</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="c15">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="c14">o</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E9:10_1_1_5_1_1"><i><font color="Green" title="E21">A53</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/>
<a NAME="E11:10_1_1_5_1_1"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/>
<b>hence </b><a NAME="E12:10_1_1_5_1_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="#E3:10_1_1_5"><i><font color="Green" title="E14">A46</font></i></a>, <a class="txt" href="#E5:10_1_1_5"><i><font color="Green" title="E15">A47</font></i></a>, <a class="txt" href="#E8:10_1_1_5"><i><font color="Green" title="E16">A48</font></i></a>, <a class="txt" href="#E4:10_1_1_5_1_1"><i><font color="Green" title="E20">A52</font></i></a>, <a class="txt" href="#E9:10_1_1_5_1_1"><i><font color="Green" title="E21">A53</font></i></a>, <a class="txt" href="#E10:10_1_1_5_1_1"><i><font color="Green" title="E22">A54</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</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>
<a NAME="E2:10_1_1_5_1"/>
<span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</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="10_1_1_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> or <font color="Maroon" title="c9">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="E1:10_1_1_5_1_2"/><i><font color="Green" title="E19">A55</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</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="c9">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="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E8:10_1_1_5"><i><font color="Green" title="E16">A48</font></i></a>, <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="c11">q1</font> = <font color="Maroon" title="c10">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/>
<b>set </b><font color="Maroon" title="c12">q2</font> = <font color="Maroon" title="c10">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E4:10_1_1_5_1_2"/><i><font color="Green" title="E20">A56</font></i>: 
<font color="Maroon" title="c10">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="10_1_1_5_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c13">q3</font> = <font color="Maroon" title="c10">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="E2:10_1_1_5_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="c13">q3</font> &amp; <font color="Maroon" title="c13">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c13">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="10_1_1_5_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_5_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="c13">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c11">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">d</font>
 
<b>by </b><i><a class="txt" href="#E1:10_1_1_5_1_2"><i><font color="Green" title="E19">A55</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">r</font>, <font color="Maroon" title="c15">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_5_1_2_1_1_1"/><i><font color="Green" title="E21">A57</font></i>: 
( <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1</font> &amp; <font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c14">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">o</font> )
 <b>by </b><i><a class="txt" href="#E8:10_1_1_5"><i><font color="Green" title="E16">A48</font></i></a></i>;<br/>
<a NAME="E4:10_1_1_5_1_2_1_1_1"/>
<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="c10">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="c10">q</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">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="E5:10_1_1_5_1_2_1_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="c13">q3</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E3:10_1_1_5"><i><font color="Green" title="E14">A46</font></i></a>, <a class="txt" href="#E8:10_1_1_5"><i><font color="Green" title="E16">A48</font></i></a>, <a class="txt" href="#E3:10_1_1_5_1_2_1_1_1"><i><font color="Green" title="E21">A57</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="E2:10_1_1_5_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="c13">q3</font> &amp; <font color="Maroon" title="c13">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c13">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="#E1:10_1_1_5"><i><font color="Green" title="E13">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</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="E3:10_1_1_5_1_2_1"/>
<font color="Maroon" title="c10">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="#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</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>
<a NAME="E5:10_1_1_5_1_2"/>
<font color="Maroon" title="c10">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="c10">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="#E1:10"><i><font color="Green" title="E11">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="E6:10_1_1_5_1_2"/>
<font color="Maroon" title="c9">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="#E4:10_1_1_5_1_2"><i><font color="Green" title="E20">A56</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="E3:10_1_1_5_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#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="#E1:10_1_1_5_1"><i><font color="Green" title="E18">A50</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="E10:10_1_1_5"/>
<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="#E5:10_1_1_5"><i><font color="Green" title="E15">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/><b>hence </b><a NAME="E11:10_1_1_5"/>
<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#V4" title="MEASURE5:attr.4">left_open_interval</a>
 <b>by </b><i><a class="txt" href="#E9:10_1_1_5"><i><font color="Green" title="E17">A49</font></i></a>, <a class="txt" href="#E8:10_1_1_5"><i><font color="Green" title="E16">A48</font></i></a>, <a class="ref" href="measure5.html#D8" title="MEASURE5:def.8">MEASURE5:def 8</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="10_1_1_6"><b>case </b></a><a NAME="E1:10_1_1_6"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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#V4" title="MEASURE5:attr.4">left_open_interval</a></span><br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_6"/>
<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#V4" title="MEASURE5:attr.4">left_open_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="E7:10"/>
<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#V4" title="MEASURE5:attr.4">left_open_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>
