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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">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#V1" title="MEASURE5:attr.1">open_interval</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:9"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">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#V1" title="MEASURE5:attr.1">open_interval</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:9"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:9"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_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="9_1_1_1"><span class="kw">case </span></a><a NAME="E1:9_1_1_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_1_1_1_1_1"/>
( ( <font color="Maroon" title="c3">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="c3">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="c3">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="c3">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="c3">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="c3">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="urysohn2.html#T6" target="_self" title="URYSOHN2:th.6">Th6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:9_1_1_1_1_1_1"/>
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:9_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:9_1_1_1_1_1_1"/><span class="kw">then </span>
<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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="urysohn2.html#T5" target="_self" title="URYSOHN2:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:9_1_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> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E2:9_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:9_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> ;<br/><a NAME="E3:9_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">s</font> is   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">d</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E5:9_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<font color="Maroon" title="c7">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="c5">s</font>
 ;<br/><a NAME="E6:9_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5: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="9_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">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/>

<span class="kw">assume </span><a NAME="E1:9_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E5:9_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<font color="Maroon" title="c9">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="9_1_1_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c9">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#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:9_1_1_1_1_1_2_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="c9">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="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E3:9_1_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E5:9_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E3:9_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:9_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c9">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="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:9_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:9_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>;<br/><a NAME="E8:9_1_1_1_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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">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="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:9_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E19">A17</font></span>: 
<font color="Maroon" title="c9">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="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:9_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E20">A18</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c10">z2</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E9:9_1_1_1_1_1_2_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r</font>, <font color="Maroon" title="c14">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:9_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E21">A19</font></span>: 
( <font color="Maroon" title="c13">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E12:9_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <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>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E14:9_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="txt" href="#E11:9_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E15:9_1_1_1_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:9_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E5:9_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="txt" href="#E11:9_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="txt" href="#E7:9_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E20">A18</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:9_1_1_1_1_1_2"/><span class="kw">then </span>
<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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E10:9_1_1_1_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:9_1_1_1_1_1_3"/>
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:9_1_1_1_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="urysohn2.html#T3" target="_self" title="URYSOHN2:th.3">Th3</a>, <a class="ref" href="xxreal_1.html#T224" title="XXREAL_1:th.224">XXREAL_1:224</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:9_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E13">A20</font></span>: 
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c4">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">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="c5">s</font>, <font color="Maroon" title="c8">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="c6">r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:9_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E14">A21</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">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="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></span>;<br/><a NAME="E5:9_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E15">A22</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5: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="9_1_1_1_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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/>

<span class="kw">assume </span><a NAME="E1:9_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E16">A23</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">set </span><font color="Maroon" title="c11">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">q1</font> = <font color="Maroon" title="c10">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E17">A24</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E16">A23</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E5:9_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E18">A25</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E16">A23</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E6:9_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E19">A26</font></span>: 
<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="9_1_1_1_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:9_1_1_1_1_1_4_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E3:9_1_1_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A27</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E17">A24</font></span></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_1_4_1_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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E5:9_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E18">A25</font></span></a>, <a class="ref" href="xreal_1.html#T77" title="XREAL_1:th.77">XREAL_1:77</a></span>;<br/>
<a NAME="E5:9_1_1_1_1_1_4_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9_1_1_1_1_1_4_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E3:9_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E20">A27</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:9_1_1_1_1_1_4_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:9_1_1_1_1_1_4_1"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E19">A26</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:9_1_1_1_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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E16">A28</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">consider </span><font color="Maroon" title="c11">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E17">A29</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:9_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E18">A30</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c11">z2</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:9_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">1o</font>, <font color="Maroon" title="c14">1ra</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:9_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E19">A31</font></span>: 
( <font color="Maroon" title="c13">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c14">1ra</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> )
 <span class="kw">and </span><br/><a NAME="E10:9_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c13">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">1ra</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E13">A20</font></span></a></span>;<br/>
<a NAME="E11:9_1_1_1_1_1_4_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">2o</font>, <font color="Maroon" title="c16">2r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E13:9_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E20">A32</font></span>: 
( <font color="Maroon" title="c15">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; <font color="Maroon" title="c16">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E14:9_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c15">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E13">A20</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">1o1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">1o</font>, <font color="Maroon" title="c18">1r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">1ra</font>, <font color="Maroon" title="c19">2o1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">2o</font>, <font color="Maroon" title="c20">2r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">2r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E16:9_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c15">2o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">2r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="txt" href="#E13:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E17:9_1_1_1_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A33</font></span>: 
<font color="Maroon" title="c20">2r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">2o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></span>;<br/>
<a NAME="E18:9_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c13">1o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">1ra</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="txt" href="#E9:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E19">A31</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E19:9_1_1_1_1_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">1r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">1o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:9_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E18">A30</font></span></a>, <a class="txt" href="#E9:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E19">A31</font></span></a>, <a class="txt" href="#E13:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="txt" href="#E17:9_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:9_1_1_1_1_1_4"/><span class="kw">then </span>
<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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E15">A22</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:9_1_1_1_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E14">A21</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1_5"><span class="kw">case </span></a><a NAME="E1:9_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E13">A34</font></span>: 
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">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="c5">s</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:9_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E14">A35</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5: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="9_1_1_1_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">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/>

<span class="kw">assume </span><a NAME="E1:9_1_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E15">A36</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E16">A37</font></span>: 
<font color="Maroon" title="c9">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="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E5:9_1_1_1_1_1_5_1"/>
<font color="Maroon" title="c9">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="9_1_1_1_1_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c9">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#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:9_1_1_1_1_1_5_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="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E15">A36</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E3:9_1_1_1_1_1_5_1_1"/><span class="kw">then </span>
<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="c9">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E15">A36</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E16">A37</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E4:9_1_1_1_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A38</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></span>;<br/>
<a NAME="E5:9_1_1_1_1_1_5_1_1"/>
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E13">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9_1_1_1_1_1_5_1_1"/>
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E17">A38</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:9_1_1_1_1_1_5_1"/>
<font color="Maroon" title="c8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E16">A37</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:9_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E15">A39</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></span>;<br/><a NAME="E6:9_1_1_1_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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">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="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E16">A40</font></span>: 
<font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E17">A41</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:9_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E18">A42</font></span>: 
<font color="Maroon" title="c9">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="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E16">A40</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">z2</font> = <font color="Maroon" title="c10">z2</font>, <font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:9_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E17">A41</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c14">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:9_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E19">A43</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> )
 <span class="kw">and </span><br/><a NAME="E10:9_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E13">A34</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E12:9_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E20">A44</font></span>: 
 <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="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></span>;<br/>
<a NAME="E13:9_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E17">A41</font></span></a>, <a class="txt" href="#E9:9_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E19">A43</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E14:9_1_1_1_1_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:9_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E18">A42</font></span></a>, <a class="txt" href="#E9:9_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E19">A43</font></span></a>, <a class="txt" href="#E12:9_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E20">A44</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:9_1_1_1_1_1_5"/><span class="kw">then </span>
<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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E14">A35</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:9_1_1_1_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E15">A39</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1_6"><span class="kw">case </span></a><a NAME="E1:9_1_1_1_1_1_6"/>
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:9_1_1_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A45</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:9_1_1_1_1_1_6"/><span class="kw">then </span>
<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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="urysohn2.html#T5" target="_self" title="URYSOHN2:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:9_1_1_1_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E2:9_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E13">A45</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:9_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2"><span class="kw">case </span></a><a NAME="E1:9_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:9_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3"><span class="kw">case </span></a><a NAME="E1:9_1_1_3"/><span class="lab"><font color="Green" title="E12">A46</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_1_1_3_1_1"/>
( ( <font color="Maroon" title="c3">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="c3">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="c3">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="c3">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="c3">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="c3">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="urysohn2.html#T6" target="_self" title="URYSOHN2:th.6">Th6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:9_1_1_3_1_1_1"/>
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:9_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A47</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:9_1_1_3_1_1_1"/><span class="kw">then </span>
<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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="urysohn2.html#T5" target="_self" title="URYSOHN2:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:9_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E2:9_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E13">A47</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:9_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E13">A48</font></span>: 
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">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="c5">s</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:9_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E14">A49</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5: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="9_1_1_3_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">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/>

<span class="kw">assume </span><a NAME="E1:9_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E15">A50</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E4:9_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E16">A51</font></span>: 
<font color="Maroon" title="c11">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E15">A50</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E5:9_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E17">A52</font></span>: 
<font color="Maroon" title="c9">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="9_1_1_3_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c9">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#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:9_1_1_3_1_1_2_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="c9">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="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E3:9_1_1_3_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A53</font></span>: 
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E16">A51</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E4:9_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E13">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E3:9_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E18">A53</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:9_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c9">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="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:9_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E17">A52</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:9_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E15">A54</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></span>;<br/><a NAME="E6:9_1_1_3_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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">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="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E16">A55</font></span>: 
<font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E17">A56</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:9_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E18">A57</font></span>: 
<font color="Maroon" title="c9">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="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">z2</font> = <font color="Maroon" title="c10">z2</font>, <font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:9_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c11">z2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E17">A56</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r</font>, <font color="Maroon" title="c14">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:9_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E19">A58</font></span>: 
( <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E10:9_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E13">A48</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <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>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E12:9_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E20">A59</font></span>: 
 <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="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></span>;<br/>
<a NAME="E13:9_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E17">A56</font></span></a>, <a class="txt" href="#E9:9_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E19">A58</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E14:9_1_1_3_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:9_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E18">A57</font></span></a>, <a class="txt" href="#E9:9_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E19">A58</font></span></a>, <a class="txt" href="#E12:9_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E20">A59</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:9_1_1_3_1_1_2"/><span class="kw">then </span>
<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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E14">A49</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:9_1_1_3_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E15">A54</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_3"><span class="kw">case </span></a><a NAME="E1:9_1_1_3_1_1_3"/>
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:9_1_1_3_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="ref" href="urysohn2.html#T3" target="_self" title="URYSOHN2:th.3">Th3</a>, <a class="ref" href="xxreal_1.html#T224" title="XXREAL_1:th.224">XXREAL_1:224</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_4"><span class="kw">case </span></a><a NAME="E1:9_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E13">A60</font></span>: 
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c4">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">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="c5">s</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">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="c6">r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><a NAME="E5:9_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E14">A61</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">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="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E6:9_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E15">A62</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5: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="9_1_1_3_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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/>

<span class="kw">assume </span><a NAME="E1:9_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E16">A63</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">set </span><font color="Maroon" title="c11">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="E3:9_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c10">q</font> is   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">q1</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E5:9_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E17">A64</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font>
 ;<br/>
<a NAME="E6:9_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E18">A65</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E16">A63</font></span></a>, <a class="txt" href="#E5:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E7:9_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E19">A66</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E16">A63</font></span></a>, <a class="txt" href="#E5:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E8:9_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E20">A67</font></span>: 
<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="9_1_1_3_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:9_1_1_3_1_1_4_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E3:9_1_1_3_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A68</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="txt" href="#E5:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="txt" href="#E7:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E19">A66</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E4:9_1_1_3_1_1_4_1_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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="txt" href="#E5:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="txt" href="#E6:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E18">A65</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E5:9_1_1_3_1_1_4_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9_1_1_3_1_1_4_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E3:9_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E21">A68</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:9_1_1_3_1_1_4_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:9_1_1_3_1_1_4_1"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E20">A67</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:9_1_1_3_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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E16">A69</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">consider </span><font color="Maroon" title="c11">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E17">A70</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:9_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E18">A71</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E16">A69</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c11">z2</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:9_1_1_3_1_1_4_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E17">A70</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">2o</font>, <font color="Maroon" title="c14">2r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:9_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E19">A72</font></span>: 
( <font color="Maroon" title="c13">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; <font color="Maroon" title="c14">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E10:9_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c13">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E13">A60</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">2o1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">2o</font>, <font color="Maroon" title="c16">2r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">2r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E12:9_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c13">2o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">2r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E17">A70</font></span></a>, <a class="txt" href="#E9:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E19">A72</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E13:9_1_1_3_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A73</font></span>: 
<font color="Maroon" title="c15">2o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">2r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E14:9_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E17">A70</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">1o</font>, <font color="Maroon" title="c18">1ra</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E16:9_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E21">A74</font></span>: 
( <font color="Maroon" title="c17">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c18">1ra</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> )
 <span class="kw">and </span><br/><a NAME="E17:9_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c17">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">1ra</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E13">A60</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">1o1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">1o</font>, <font color="Maroon" title="c20">1r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">1ra</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E19:9_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c17">1o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">1ra</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E17">A70</font></span></a>, <a class="txt" href="#E16:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E21">A74</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E20:9_1_1_3_1_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">1o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">1r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:9_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E18">A71</font></span></a>, <a class="txt" href="#E16:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E21">A74</font></span></a>, <a class="txt" href="#E9:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E19">A72</font></span></a>, <a class="txt" href="#E13:9_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E20">A73</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:9_1_1_3_1_1_4"/><span class="kw">then </span>
<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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E15">A62</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E9:9_1_1_3_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E14">A61</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_5"><span class="kw">case </span></a><a NAME="E1:9_1_1_3_1_1_5"/><span class="lab"><font color="Green" title="E13">A75</font></span>: 
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">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="c5">s</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:9_1_1_3_1_1_5"/><span class="lab"><font color="Green" title="E14">A76</font></span>: 
<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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">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="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E15">A77</font></span>: 
<font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:9_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E16">A78</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:9_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E17">A79</font></span>: 
<font color="Maroon" title="c9">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="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E15">A77</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:9_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E18">A80</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c10">z2</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E9:9_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E16">A78</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c14">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:9_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E19">A81</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> )
 <span class="kw">and </span><br/><a NAME="E12:9_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_5"><span class="lab"><font color="Green" title="E13">A75</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">r</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E14:9_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E16">A78</font></span></a>, <a class="txt" href="#E11:9_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E19">A81</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E15:9_1_1_3_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:9_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E17">A79</font></span></a>, <a class="txt" href="#E11:9_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E19">A81</font></span></a>, <a class="txt" href="#E7:9_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E18">A80</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:9_1_1_3_1_1_5"/><span class="lab"><font color="Green" title="E15">A82</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>;<br/><a NAME="E6:9_1_1_3_1_1_5"/>
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5: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="9_1_1_3_1_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">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/>

<span class="kw">assume </span><a NAME="E1:9_1_1_3_1_1_5_2"/><span class="lab"><font color="Green" title="E16">A83</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E4:9_1_1_3_1_1_5_2"/><span class="lab"><font color="Green" title="E17">A84</font></span>: 
<font color="Maroon" title="c9">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="c9">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E5:9_1_1_3_1_1_5_2"/><span class="lab"><font color="Green" title="E18">A85</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E16">A83</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>;<br/>
<a NAME="E6:9_1_1_3_1_1_5_2"/>
<font color="Maroon" title="c9">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="9_1_1_3_1_1_5_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c9">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#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:9_1_1_3_1_1_5_2_1"/>
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font> &amp; <font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="txt" href="#E1:9_1_1_3_1_1_5"><span class="lab"><font color="Green" title="E13">A75</font></span></a>, <a class="txt" href="#E5:9_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E18">A85</font></span></a>, <a class="txt" href="#E4:9_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E17">A84</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:9_1_1_3_1_1_5_2_1"/>
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:9_1_1_3_1_1_5_2"/>
<font color="Maroon" title="c8">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E17">A84</font></span></a>, <a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:9_1_1_3_1_1_5"/><span class="kw">then </span>
<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="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_3_1_1_5"><span class="lab"><font color="Green" title="E14">A76</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:9_1_1_3_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_3_1_1_5"><span class="lab"><font color="Green" title="E15">A82</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_6"><span class="kw">case </span></a><a NAME="E1:9_1_1_3_1_1_6"/>
( <font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:9_1_1_3_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A86</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:9_1_1_3_1_1_6"/><span class="kw">then </span>
<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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="urysohn2.html#T5" target="_self" title="URYSOHN2:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:9_1_1_3_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E2:9_1_1_3_1_1_6"><span class="lab"><font color="Green" title="E13">A86</font></span></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:9_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#V1" title="MEASURE5:attr.1">open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:9"/>
<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#V1" title="MEASURE5:attr.1">open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
