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<b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>given </b><font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">b1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><a NAME="E1:60_1_2"/><i><font color="Green" title="E53">A2</font></i>: 
( ( <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="c4">a1</font>,<font color="Maroon" title="c5">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#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">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="c4">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</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="c4">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) &amp; ( <font color="Maroon" title="c4">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b1</font> implies <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">a1</font> ) &amp; ( <font color="Maroon" title="c5">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font> implies <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 ;<br/>

<b>given </b><font color="Maroon" title="c6">a2</font>, <font color="Maroon" title="c7">b2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><a NAME="E2:60_1_2"/><i><font color="Green" title="E54">A3</font></i>: 
( ( <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="c6">a2</font>,<font color="Maroon" title="c7">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#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a2</font>,<font color="Maroon" title="c7">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="c6">a2</font>,<font color="Maroon" title="c7">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</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="c6">a2</font>,<font color="Maroon" title="c7">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) &amp; ( <font color="Maroon" title="c6">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b2</font> implies <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">a2</font> ) &amp; ( <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a2</font> implies <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_2_1"/>
( ( <font color="Maroon" title="c4">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b2</font> ) or ( <font color="Maroon" title="c4">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a2</font> ) or ( <font color="Maroon" title="c5">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b2</font> ) or ( <font color="Maroon" title="c5">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1"><b>suppose </b></a><a NAME="E1:60_1_2_1_1"/>
( <font color="Maroon" title="c4">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b2</font> )
 ;<br/><div class="add"><a NAME="E2:60_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:60_1_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="measure5.html#E2:60_1_2"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="measure5.html#T52" target="_self" title="MEASURE5:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:60_1_2_1_1"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:60_1_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="measure5.html#E2:60_1_2"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_2"><b>suppose </b></a><a NAME="E1:60_1_2_1_2"/><i><font color="Green" title="E55">A4</font></i>: 
( <font color="Maroon" title="c4">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a2</font> )
 ;<br/><div class="add"><a NAME="E2:60_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:60_1_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="measure5.html#E2:60_1_2"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="measure5.html#T52" target="_self" title="MEASURE5:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:60_1_2_1_2"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:60_1_2_1_2"><i><font color="Green" title="E55">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_3"><b>suppose </b></a><a NAME="E1:60_1_2_1_3"/><i><font color="Green" title="E55">A5</font></i>: 
( <font color="Maroon" title="c5">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c6">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b2</font> )
 ;<br/><div class="add"><a NAME="E2:60_1_2_1_3"/><b>then </b>
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a2</font> &amp; <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:60_1_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="measure5.html#E2:60_1_2"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="measure5.html#T52" target="_self" title="MEASURE5:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:60_1_2_1_3"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:60_1_2_1_3"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_4"><b>suppose </b></a><a NAME="E1:60_1_2_1_4"/>
( <font color="Maroon" title="c5">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_2_1_4"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:60_1_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="measure5.html#E2:60_1_2"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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