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<b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44_1_2"/><i><font color="Green" title="E31">A2</font></i>: 
 ex <font color="Olive" title="b1">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Maroon" title="c2">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font> &amp; ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> or <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="c2">a1</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>and </b><br/><a NAME="E2:44_1_2"/><i><font color="Green" title="E32">A3</font></i>: 
 ex <font color="Olive" title="b1">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font> &amp; ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> or <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">a2</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Olive" title="b1">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">b1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E4:44_1_2"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b1</font> &amp; ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> or <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="c2">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure6.html#E1:44_1_2"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">b2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E6:44_1_2"/><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">b2</font> &amp; ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> or <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">a2</font>,<font color="Maroon" title="c5">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure6.html#E2:44_1_2"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_2_1"/>
( ( <font color="Maroon" title="c2">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b2</font> ) or ( <font color="Maroon" title="c2">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> ) or ( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b2</font> ) or ( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> ) )
 </span><b>by </b><i><a class="txt" href="measure6.html#E4:44_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure6.html#E6:44_1_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_1"><b>suppose </b></a><a NAME="E1:44_1_2_1_1"/>
( <font color="Maroon" title="c2">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_2_1_1"/>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="txt" href="measure6.html#E4:44_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure6.html#E6:44_1_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="measure5.html#T52" title="MEASURE5:th.52">MEASURE5:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_2"><b>suppose </b></a><a NAME="E1:44_1_2_1_2"/>
( <font color="Maroon" title="c2">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_2_1_2"/>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="txt" href="measure6.html#E4:44_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure6.html#E6:44_1_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="measure5.html#T52" title="MEASURE5:th.52">MEASURE5:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_3"><b>suppose </b></a><a NAME="E1:44_1_2_1_3"/>
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_2_1_3"/>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="txt" href="measure6.html#E4:44_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure6.html#E6:44_1_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="measure5.html#T52" title="MEASURE5:th.52">MEASURE5:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_4"><b>suppose </b></a><a NAME="E1:44_1_2_1_4"/><i><font color="Green" title="E35">A6</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> )
 ;<br/><div class="add"><a NAME="E2:44_1_2_1_4"/><b>then </b>
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:44"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="measure6.html#E4:44_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure6.html#E6:44_1_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="measure5.html#T13" title="MEASURE5:th.13">MEASURE5:13</a>, <a class="ref" href="measure5.html#T14" title="MEASURE5:th.14">MEASURE5:14</a></i>;<br/><a NAME="E3:44_1_2_1_4"/><b>then </b>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:44"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="measure6.html#E4:44_1_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure6.html#E6:44_1_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="measure6.html#E1:44_1_2_1_4"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="measure5.html#T13" title="MEASURE5:th.13">MEASURE5:13</a>, <a class="ref" href="measure5.html#T14" title="MEASURE5:th.14">MEASURE5:14</a></i>;<br/><b>hence </b><a NAME="E4:44_1_2_1_4"/>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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