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<div class="add">

<b>thus </b><a NAME="E1:12_1_1"/>
(  not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</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">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> ) implies <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green" title="E5">Z</font></i>: 
<font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</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">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_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="E1:12_1_1_1_1"/>
( ( <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> ) or ( not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> ) or ( <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> ) or ( not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_1"/><i><font color="Green" title="E6">S</font></i>: 
( <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</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">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c2">a</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font>, <font color="Maroon" title="c3">b</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/><a NAME="E3:12_1_1_1_1_1"/><i><font color="Green" title="E7">W</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font> &amp; <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1_1_1"><i><font color="Green" title="E6">S</font></i></a>, <a class="txt" href="#E1:12_1_1_1"><i><font color="Green" title="E5">Z</font></i></a>, <a class="ref" href="xxreal_2.html#T79" title="XXREAL_2:th.79">XXREAL_2:79</a>, <a class="ref" href="xxreal_2.html#D5" title="XXREAL_2:def.5">XXREAL_2:def 5</a>, <a class="ref" href="xxreal_2.html#D6" title="XXREAL_2:def.6">XXREAL_2:def 6</a></i>;<br/><a NAME="E4:12_1_1_1_1_1"/><i><font color="Green" title="E8">X</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_1_1_1"><i><font color="Green" title="E7">W</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E5:12_1_1_1_1_1"/>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_1_1_1"><i><font color="Green" title="E7">W</font></i></a>, <a class="ref" href="measure5.html#D6" target="_self" title="MEASURE5:def.6">Def6</a>, <a class="txt" href="#E4:12_1_1_1_1_1"><i><font color="Green" title="E8">X</font></i></a></i>;<br/><b>hence </b><a NAME="E6:12_1_1_1_1_1"/>
( <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 ; <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="12_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_2"/><i><font color="Green" title="E6">S</font></i>: 
( not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</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">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )</span><br/><div class="add"><a NAME="E2:12_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E7">X</font></i>: 
<font color="Maroon" title="c1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font></span>)</span></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1"><i><font color="Green" title="E5">Z</font></i></a>, <a class="ref" href="xxreal_2.html#T80" title="XXREAL_2:th.80">XXREAL_2:80</a></i>;<br/><b>set </b><font color="Maroon" title="c2">a</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font>;<br/><b>set </b><font color="Maroon" title="c3">b</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font>;<br/><a NAME="E3:12_1_1_1_1_2"/><i><font color="Green" title="E8">Y</font></i>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1_1_2"><i><font color="Green" title="E6">S</font></i></a>, <a class="ref" href="xxreal_2.html#D6" title="XXREAL_2:def.6">XXREAL_2:def 6</a></i>;<br/><a NAME="E4:12_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E9">W</font></i>: 
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font> &amp; <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font></span>)</span></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> )
 <b>by </b><i><a class="txt" href="#E2:12_1_1_1_1_2"><i><font color="Green" title="E7">X</font></i></a>, <a class="ref" href="xxreal_2.html#T40" title="XXREAL_2:th.40">XXREAL_2:40</a></i>;<br/><a NAME="E5:12_1_1_1_1_2"/>
not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1_1_2"><i><font color="Green" title="E6">S</font></i></a>, <a class="ref" href="xxreal_2.html#D5" title="XXREAL_2:def.5">XXREAL_2:def 5</a></i>;<br/><a NAME="E6:12_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E10">WW</font></i>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E4:12_1_1_1_1_2"><i><font color="Green" title="E9">W</font></i></a>, <a class="txt" href="#E3:12_1_1_1_1_2"><i><font color="Green" title="E8">Y</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">a</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font>, <font color="Maroon" title="c5">b</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/><a NAME="E8:12_1_1_1_1_2"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E6:12_1_1_1_1_2"><i><font color="Green" title="E10">WW</font></i></a>, <a class="txt" href="#E4:12_1_1_1_1_2"><i><font color="Green" title="E9">W</font></i></a></i>;<br/><a NAME="E9:12_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">b</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> 
 ;<br/><a NAME="E10:12_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a>
 <b>by </b><i><a class="txt" href="#E4:12_1_1_1_1_2"><i><font color="Green" title="E9">W</font></i></a>, <a class="ref" href="measure5.html#D8" target="_self" title="MEASURE5:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E11:12_1_1_1_1_2"/>
( <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 ; <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="12_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_3"/><i><font color="Green" title="E6">S</font></i>: 
( <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</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">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )</span><br/><div class="add"><a NAME="E2:12_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E7">X</font></i>: 
<font color="Maroon" title="c1">IT</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font></span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1"><i><font color="Green" title="E5">Z</font></i></a>, <a class="ref" href="xxreal_2.html#T81" title="XXREAL_2:th.81">XXREAL_2:81</a></i>;<br/><b>set </b><font color="Maroon" title="c2">a</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font>;<br/><b>set </b><font color="Maroon" title="c3">b</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font>;<br/><a NAME="E3:12_1_1_1_1_3"/><i><font color="Green" title="E8">Y</font></i>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1_1_3"><i><font color="Green" title="E6">S</font></i></a>, <a class="ref" href="xxreal_2.html#D5" title="XXREAL_2:def.5">XXREAL_2:def 5</a></i>;<br/><a NAME="E4:12_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E9">W</font></i>: 
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font> &amp; <font color="Maroon" title="c1">IT</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font></span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )
 <b>by </b><i><a class="txt" href="#E2:12_1_1_1_1_3"><i><font color="Green" title="E7">X</font></i></a>, <a class="ref" href="xxreal_2.html#T40" title="XXREAL_2:th.40">XXREAL_2:40</a></i>;<br/><a NAME="E5:12_1_1_1_1_3"/>
not  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1_1_3"><i><font color="Green" title="E6">S</font></i></a>, <a class="ref" href="xxreal_2.html#D6" title="XXREAL_2:def.6">XXREAL_2:def 6</a></i>;<br/><a NAME="E6:12_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E10">WW</font></i>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E4:12_1_1_1_1_3"><i><font color="Green" title="E9">W</font></i></a>, <a class="txt" href="#E3:12_1_1_1_1_3"><i><font color="Green" title="E8">Y</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">a</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">IT</font>, <font color="Maroon" title="c5">b</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">IT</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/><a NAME="E8:12_1_1_1_1_3"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="txt" href="#E6:12_1_1_1_1_3"><i><font color="Green" title="E10">WW</font></i></a>, <a class="txt" href="#E4:12_1_1_1_1_3"><i><font color="Green" title="E9">W</font></i></a></i>;<br/><a NAME="E9:12_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">a</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> 
 ;<br/><a NAME="E10:12_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>
 <b>by </b><i><a class="txt" href="#E4:12_1_1_1_1_3"><i><font color="Green" title="E9">W</font></i></a>, <a class="ref" href="measure5.html#D7" target="_self" title="MEASURE5:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E11:12_1_1_1_1_3"/>
( <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 ; <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="12_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_4"/>
( not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; not <font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</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">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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="E3:12_1_1_1_1_4"/><i><font color="Green" title="E6">W</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <b>by </b><i><a class="txt" href="#E1:12_1_1_1"><i><font color="Green" title="E5">Z</font></i></a>, <a class="ref" href="xxreal_2.html#T83" title="XXREAL_2:th.83">XXREAL_2:83</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c3">b</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/><a NAME="E5:12_1_1_1_1_4"/>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_1_1_4"><i><font color="Green" title="E6">W</font></i></a></i>;<br/><a NAME="E6:12_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_1_1_4"><i><font color="Green" title="E6">W</font></i></a>, <a class="ref" href="measure5.html#D5" target="_self" title="MEASURE5:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E7:12_1_1_1_1_4"/>
( <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 ; <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>assume </b><a NAME="E2:12_1_1"/><i><font color="Green" title="E5">Z</font></i>: 
( <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</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">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_2"/>
( <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> or <font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> )
 </span><b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E5">Z</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1"><b>suppose </b></a><a NAME="E1:12_1_1_2_1"/>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a></span><br/><div class="add"><a NAME="E2:12_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">IT</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self" title="MEASURE5:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_2_1"/>
<font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>
 ; <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="12_1_1_2_2"><b>suppose </b></a><a NAME="E1:12_1_1_2_2"/>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</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">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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> <b> such that </b><br/><a NAME="E3:12_1_1_2_2"/><i><font color="Green" title="E6">W</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self" title="MEASURE5:def.6">Def6</a></i>;<br/><b>thus </b><a NAME="E4:12_1_1_2_2"/>
<font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_2_2"><i><font color="Green" title="E6">W</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_3"><b>suppose </b></a><a NAME="E1:12_1_1_2_3"/>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</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">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">a</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>, <font color="Maroon" title="c3">b</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> <b> such that </b><br/><a NAME="E3:12_1_1_2_3"/><i><font color="Green" title="E6">W</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self" title="MEASURE5:def.8">Def8</a></i>;<br/><b>thus </b><a NAME="E4:12_1_1_2_3"/>
<font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_2_3"><i><font color="Green" title="E6">W</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_4"><b>suppose </b></a><a NAME="E1:12_1_1_2_4"/>
<font color="Maroon" title="c1">IT</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</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">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">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="c3">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:12_1_1_2_4"/><i><font color="Green" title="E6">W</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self" title="MEASURE5:def.7">Def7</a></i>;<br/><b>thus </b><a NAME="E4:12_1_1_2_4"/>
<font color="Maroon" title="c1">IT</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_2_4"><i><font color="Green" title="E6">W</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/></div><b>end;</b></div></div><b>end;</b></div>

</div>
