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<div class="add">

<span class="kw">thus </span><a NAME="E1:19_1_1"/>
(  not <font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="measure5.html#NV7" title="MEASURE5:NV.7">interval</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="19_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_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="19_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_1_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">reconsider </span><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#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/><a NAME="E3:19_1_1_1_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
( <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="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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:19_1_1_1_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></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></span>;<br/><a NAME="E4:19_1_1_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:19_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E3:19_1_1_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="measure5.html#D6" target="_self" title="MEASURE5:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_1_1_2"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
( 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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">set </span><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/><span class="kw">set </span><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="E2:19_1_1_1_1_2"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:19_1_1_1_1_2"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="xxreal_2.html#T80" title="XXREAL_2:th.80">XXREAL_2:80</a></span>;<br/><a NAME="E3:19_1_1_1_1_2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1_1_2"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="xxreal_2.html#D6" title="XXREAL_2:def.6">XXREAL_2:def 6</a></span>;<br/><a NAME="E4:19_1_1_1_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T40" title="XXREAL_2:th.40">XXREAL_2:40</a></span>;<br/><span class="kw">hence </span><a NAME="E5:19_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E2:19_1_1_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="measure5.html#D8" target="_self" title="MEASURE5:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_1_1_3"/><span class="lab"><font color="Green" title="E6">A7</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">set </span><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/><span class="kw">set </span><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="E2:19_1_1_1_1_3"/><span class="lab"><font color="Green" title="E7">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:19_1_1_1_1_3"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="xxreal_2.html#T81" title="XXREAL_2:th.81">XXREAL_2:81</a></span>;<br/><a NAME="E3:19_1_1_1_1_3"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1_1_3"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="xxreal_2.html#D5" title="XXREAL_2:def.5">XXREAL_2:def 5</a></span>;<br/><a NAME="E4:19_1_1_1_1_3"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T40" title="XXREAL_2:th.40">XXREAL_2:40</a></span>;<br/><span class="kw">hence </span><a NAME="E5:19_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E2:19_1_1_1_1_3"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="ref" href="measure5.html#D7" target="_self" title="MEASURE5:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:19_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:19_1_1_1_1_4"/><span class="lab"><font color="Green" title="E6">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:19_1_1_1_1_4"/><span class="lab"><font color="Green" title="E7">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="xxreal_2.html#T83" title="XXREAL_2:th.83">XXREAL_2:83</a></span>;<br/><span class="kw">reconsider </span><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#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><a NAME="E6:19_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1_1_1_4"><span class="lab"><font color="Green" title="E6">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:19_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_1_1_4"><span class="lab"><font color="Green" title="E7">A11</font></span></a>, <a class="ref" href="measure5.html#D5" target="_self" title="MEASURE5:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E5">A12</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E5">A12</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:19_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> </span><br/><div class="add"><a NAME="E2:19_1_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D5" target="_self" title="MEASURE5:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_2_1"/>
<font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:19_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> </span><br/><div class="add"><a NAME="E2:19_1_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">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>  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="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D6" target="_self" title="MEASURE5:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_2_2"/>
<font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2_3"><span class="kw">suppose </span></a><a NAME="E1:19_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> </span><br/><div class="add"><a NAME="E2:19_1_1_2_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ex <font color="Olive" title="b2">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>  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="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D8" target="_self" title="MEASURE5:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_2_3"/>
<font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2_4"><span class="kw">suppose </span></a><a NAME="E1:19_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> </span><br/><div class="add"><a NAME="E2:19_1_1_2_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  ex <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">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="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D7" target="_self" title="MEASURE5:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_2_4"/>
<font color="Maroon" title="c1">IT</font> is  <a href="measure5.html#NV7" title="MEASURE5:NV.7">interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
