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<a NAME="E1:60_1_1"/>
( <font color="Maroon">A</font> is <a href="measure5.html#V1" target="_self">open_interval</a> or <font color="Maroon">A</font> is <a href="measure5.html#V2" target="_self">closed_interval</a> or <font color="Maroon">A</font> is <a href="measure5.html#V3" target="_self">right_open_interval</a> or <font color="Maroon">A</font> is <a href="measure5.html#V4" target="_self">left_open_interval</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:60_1_1"/><b>then </b>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:60_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>
<a NAME="E5:60_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> implies  ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> implies <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a</font> ) &amp; ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> implies <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) ) )
 
;<br/>
<a NAME="E6:60_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> implies <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a</font> ) &amp; ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> implies <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 
;<br/>
<b>hence </b><a NAME="E7:60_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) &amp; ( <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="supinf_2.html#K4">-</a> <font color="Olive">a</font> ) &amp; ( <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 <b>by </b><i/>;<br/>


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