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<b>let </b><font color="Maroon" title="c1">b1</font>, <font color="Maroon" title="c2">b2</font>, <font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b2</font> &amp; ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">b1</font> or <font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b2</font> ) implies  ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) ) )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b2</font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">b1</font> or <font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">b1</font> or <font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:30"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>case </b></a><a NAME="E1:30_1_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_1"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">a2</font> or <font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_1"><b>case </b></a><a NAME="E1:30_1_1_1_1_1"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">a2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span></span><br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_1_1_1"/>
not <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="measure5.html#E1:30_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xxreal_1.html#T112" title="XXREAL_1:th.112">XXREAL_1:112</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_2"><b>case </b></a><a NAME="E1:30_1_1_1_1_2"/>
<font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span></span><br/><div class="add"><a NAME="E2:30_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">b1</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:30_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:30_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:30"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:30_1_1_1_1_2"/>
not <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="measure5.html#E1:30"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xxreal_1.html#T127" title="XXREAL_1:th.127">XXREAL_1:127</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2"><b>case </b></a><a NAME="E1:30_1_1_2"/>
<font color="Maroon" title="c4">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span></span><br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_2"/>
not <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="measure5.html#E1:30"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xxreal_1.html#T127" title="XXREAL_1:th.127">XXREAL_1:127</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">x</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:30"/><i><font color="Green" title="E24">A4</font></i>: 
( ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )</span><br/>

<b>thus </b><a NAME="E7:30"/>
( ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E5:30"><i><font color="Green" title="E24">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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