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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  <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="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_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"> (  <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> &amp; <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>  implies <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E10">A1</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"> (  not <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>  or <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:11"/>
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V3" title="MEASURE5:attr.3">right_open_interval</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> , <font color="Maroon" title="c4">b</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E4:11"/><span class="lab"><font color="Green" title="E11">A2</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="E5:11"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D7" title="MEASURE5:def.7">MEASURE5:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_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="E1:11_1_1"/>
( ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E11">A2</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="11_1_1_1"><span class="kw">case </span></a><a NAME="E1:11_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V3" title="MEASURE5:attr.3">right_open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_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#V3" title="MEASURE5:attr.3">right_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="11_1_1_2"><span class="kw">case </span></a><a NAME="E1:11_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V3" title="MEASURE5:attr.3">right_open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_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#V3" title="MEASURE5:attr.3">right_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="11_1_1_3"><span class="kw">case </span></a><a NAME="E1:11_1_1_3"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V3" title="MEASURE5:attr.3">right_open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_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#V3" title="MEASURE5:attr.3">right_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="11_1_1_4"><span class="kw">case </span></a><a NAME="E1:11_1_1_4"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V3" title="MEASURE5:attr.3">right_open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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="E3:11_1_1_4"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ;<br/><a NAME="E4:11_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">r</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">g</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="E6:11_1_1_4"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">r</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c8">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E8:11_1_1_4"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_4"><span class="lab"><font color="Green" title="E13">A4</font></span></a></span>;<br/><a NAME="E9:11_1_1_4"/><span class="lab"><font color="Green" title="E17">A8</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="c8">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="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E8:11_1_1_4"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E3:11_1_1_4"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E10:11_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">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="c9">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="E12:11_1_1_4"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">s</font>
 ;<br/><a NAME="E13:11_1_1_4"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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="11_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_1_4_1"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11_1_1_4"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">q2</font> = <font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E3:11_1_1_4_1"/>
<font color="Maroon" title="c11">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="c13">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="E5:11_1_1_4_1"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<font color="Maroon" title="c13">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font>
 ;<br/>
<a NAME="E6:11_1_1_4_1"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:11_1_1_4_1_1"/>
<font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> is   <a href="supinf_1.html#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="c14">q3</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="E3:11_1_1_4_1_1"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c14">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 ;<br/>
<a NAME="E4:11_1_1_4_1_1"/><span class="lab"><font color="Green" title="E24">A15</font></span>: 
<font color="Maroon" title="c14">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_4_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:11_1_1_4_1_1_1"/>
<font color="Maroon" title="c13">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_4_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E5:11_1_1_4_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">p</font>, <font color="Maroon" title="c16">o</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:11_1_1_4_1_1_1"/><span class="lab"><font color="Green" title="E25">A16</font></span>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q1</font> &amp; <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> )
 <span class="kw">and </span><br/><a NAME="E4:11_1_1_4_1_1_1"/>
<font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1_1_4"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E5:11_1_1_4_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a></span>;<br/>
<a NAME="E5:11_1_1_4_1_1_1"/>
<font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_4_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E5:11_1_1_4_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E3:11_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<a NAME="E6:11_1_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">o</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:11_1_1_4_1_1_1"/>
<font color="Maroon" title="c14">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:11_1_1_4"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:11_1_1_4"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E5:11_1_1_4_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E3:11_1_1_4_1_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="txt" href="#E3:11_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</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="E5:11_1_1_4_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_4_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:11_1_1_4_1_1_2"/>
( <font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">q1</font> &amp; <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E1:11_1_1_4_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E5:11_1_1_4_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:11_1_1_4_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">q3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E8:11_1_1_4"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E12:11_1_1_4"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E5:11_1_1_4_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E3:11_1_1_4_1_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</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="E6:11_1_1_4_1_1"/>
<font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E3:11_1_1_4_1_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="txt" href="#E4:11_1_1_4_1_1"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</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="E7:11_1_1_4_1"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">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="E8:11_1_1_4_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1_1_4_1"><span class="lab"><font color="Green" title="E22">A13</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="E14:11_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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_1_4_2"/><span class="lab"><font color="Green" title="E20">A17</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">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="E4:11_1_1_4_2"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c12">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:11_1_1_4_2"/><span class="lab"><font color="Green" title="E22">A19</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_4_2"><span class="lab"><font color="Green" title="E20">A17</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="c13">z2</font> = <font color="Maroon" title="c12">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="E7:11_1_1_4_2"/>
<font color="Maroon" title="c13">z2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:11_1_1_4_2"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">2o</font>, <font color="Maroon" title="c15">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="E9:11_1_1_4_2"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
( <font color="Maroon" title="c14">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z2</font> &amp; <font color="Maroon" title="c15">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="E10:11_1_1_4_2"/>
<font color="Maroon" title="c14">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_4"><span class="lab"><font color="Green" title="E13">A4</font></span></a></span>;<br/>
<a NAME="E11:11_1_1_4_2"/>
( <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">2o</font> is   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> &amp; <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">2r</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="c16">2o1</font>, <font color="Maroon" title="c17">2r1</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="E13:11_1_1_4_2"/><span class="lab"><font color="Green" title="E24">A21</font></span>: 
( <font color="Maroon" title="c16">2o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">2o</font> &amp; <font color="Maroon" title="c17">2r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">2r</font> )
 ;<br/>
<a NAME="E14:11_1_1_4_2"/>
<font color="Maroon" title="c14">2o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">2r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:11_1_1_4_2"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="txt" href="#E9:11_1_1_4_2"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<a NAME="E15:11_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A22</font></span>: 
<font color="Maroon" title="c16">2o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">2r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E13:11_1_1_4_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E16:11_1_1_4_2"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:11_1_1_4_2"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">1o</font>, <font color="Maroon" title="c19">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="E18:11_1_1_4_2"/><span class="lab"><font color="Green" title="E26">A23</font></span>: 
( <font color="Maroon" title="c18">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c19">1ra</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z2</font> )
 <span class="kw">and </span><br/><a NAME="E19:11_1_1_4_2"/><span class="lab"><font color="Green" title="E27">A24</font></span>: 
<font color="Maroon" title="c18">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">1ra</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_4"><span class="lab"><font color="Green" title="E13">A4</font></span></a></span>;<br/>
<a NAME="E20:11_1_1_4_2"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">1o</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="c19">1ra</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E19:11_1_1_4_2"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:11_1_1_4_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1_1_4"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E3:11_1_1_4"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E12:11_1_1_4"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E6:11_1_1_4"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E5:11_1_1_4_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="txt" href="#E18:11_1_1_4_2"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E9:11_1_1_4_2"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="txt" href="#E13:11_1_1_4_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E15:11_1_1_4_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</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="E15:11_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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">g</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1_1_4"><span class="lab"><font color="Green" title="E19">A10</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="E16:11_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#V3" title="MEASURE5:attr.3">right_open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11_1_1_4"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E6:11_1_1_4"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E9:11_1_1_4"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="measure5.html#D7" title="MEASURE5:def.7">MEASURE5:def 7</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="11_1_1_5"><span class="kw">case </span></a><a NAME="E1:11_1_1_5"/><span class="lab"><font color="Green" title="E13">A25</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V3" title="MEASURE5:attr.3">right_open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:11_1_1_5"/><span class="lab"><font color="Green" title="E14">A26</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 ;<br/><a NAME="E4:11_1_1_5"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">s</font> is   <a href="supinf_1.html#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="E6:11_1_1_5"/><span class="lab"><font color="Green" title="E15">A27</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="c6">s</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c8">c</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="E8:11_1_1_5"/><span class="lab"><font color="Green" title="E16">A28</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><a NAME="E9:11_1_1_5"/><span class="lab"><font color="Green" title="E17">A29</font></span>: 
<span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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="11_1_1_5_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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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="E1:11_1_1_5_1"/><span class="lab"><font color="Green" title="E18">A30</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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1_1_5"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<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="E3:11_1_1_5_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="E5:11_1_1_5_1"/><span class="lab"><font color="Green" title="E19">A31</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="E6:11_1_1_5_1"/><span class="lab"><font color="Green" title="E20">A32</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="11_1_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:11_1_1_5_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> 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="c13">q3</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="E3:11_1_1_5_1_1"/><span class="lab"><font color="Green" title="E21">A33</font></span>: 
<font color="Maroon" title="c13">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">x</font>
 ;<br/>
<a NAME="E4:11_1_1_5_1_1"/><span class="lab"><font color="Green" title="E22">A34</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:11_1_1_5_1_1_1"/>
( <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">q1</font> &amp; <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="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E1:11_1_1_5_1"><span class="lab"><font color="Green" title="E18">A30</font></span></a>, <a class="txt" href="#E5:11_1_1_5_1"><span class="lab"><font color="Green" title="E19">A31</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:11_1_1_5_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">q3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:11_1_1_5"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E6:11_1_1_5"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="#E5:11_1_1_5_1"><span class="lab"><font color="Green" title="E19">A31</font></span></a>, <a class="txt" href="#E3:11_1_1_5_1_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</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="E5:11_1_1_5_1_1"/>
<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="#E1:11_1_1_5"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E3:11_1_1_5_1_1"><span class="lab"><font color="Green" title="E21">A33</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="E6:11_1_1_5_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="#E5:11"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E3:11_1_1_5_1_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="txt" href="#E4:11_1_1_5_1_1"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</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="E7:11_1_1_5_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="#E1:11"><span class="lab"><font color="Green" title="E10">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="E8:11_1_1_5_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="#E6:11_1_1_5_1"><span class="lab"><font color="Green" title="E20">A32</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="E10:11_1_1_5"/><span class="lab"><font color="Green" title="E18">A35</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1_1_5"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>;<br/><a NAME="E11:11_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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_5_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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_1_5_2"/><span class="lab"><font color="Green" title="E19">A36</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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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="E4:11_1_1_5_2"/><span class="lab"><font color="Green" title="E20">A37</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="E5:11_1_1_5_2"/><span class="lab"><font color="Green" title="E21">A38</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="#E1:11_1_1_5_2"><span class="lab"><font color="Green" title="E19">A36</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">q</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="E7:11_1_1_5_2"/><span class="lab"><font color="Green" title="E22">A39</font></span>: 
<font color="Maroon" title="c12">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="c13">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="E9:11_1_1_5_2"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:11_1_1_5_2"><span class="lab"><font color="Green" title="E20">A37</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">o</font>, <font color="Maroon" title="c15">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:11_1_1_5_2"/><span class="lab"><font color="Green" title="E23">A40</font></span>: 
( <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z2</font> )
 <span class="kw">and </span><br/><a NAME="E12:11_1_1_5_2"/><span class="lab"><font color="Green" title="E24">A41</font></span>: 
<font color="Maroon" title="c14">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_5"><span class="lab"><font color="Green" title="E13">A25</font></span></a></span>;<br/>
<a NAME="E13:11_1_1_5_2"/>
<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> <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="c15">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E12:11_1_1_5_2"><span class="lab"><font color="Green" title="E24">A41</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:11_1_1_5_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_5"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E8:11_1_1_5"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="txt" href="#E6:11_1_1_5"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="#E5:11_1_1_5_2"><span class="lab"><font color="Green" title="E21">A38</font></span></a>, <a class="txt" href="#E11:11_1_1_5_2"><span class="lab"><font color="Green" title="E23">A40</font></span></a>, <a class="txt" href="#E7:11_1_1_5_2"><span class="lab"><font color="Green" title="E22">A39</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</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="E12:11_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="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">c</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11_1_1_5"><span class="lab"><font color="Green" title="E17">A29</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="E13:11_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#V3" title="MEASURE5:attr.3">right_open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1_1_5"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="#E10:11_1_1_5"><span class="lab"><font color="Green" title="E18">A35</font></span></a>, <a class="ref" href="measure5.html#D7" title="MEASURE5:def.7">MEASURE5:def 7</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="11_1_1_6"><span class="kw">case </span></a><a NAME="E1:11_1_1_6"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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#V3" title="MEASURE5:attr.3">right_open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:11_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#V3" title="MEASURE5:attr.3">right_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="E8:11"/>
<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#V3" title="MEASURE5:attr.3">right_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>
