<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</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>  holds <br/> ( <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  iff <font color="Olive" title="b1">x</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  iff <font color="Maroon" title="c2">x</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )</span><br/>



<a NAME="E1:25"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a><br/>  for <font color="Olive" title="b2">y</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 <font color="Olive" title="b1">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  holds <br/><font color="Olive" title="b2">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Olive" title="b1">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</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 <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  holds <br/><font color="Olive" title="b1">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">y</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  implies <font color="Maroon" title="c4">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">y</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/>
<span class="kw">assume </span><a NAME="E3:25_1"/>
<font color="Maroon" title="c3">B</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</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> <span class="kw"> such that </span><br/><a NAME="E5:25_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c3">B</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="c7">a1</font>,<font color="Maroon" title="c8">b1</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#D3" title="MEASURE5:def.3">MEASURE5:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">b1</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c12">t</font> = <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c10">b</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<a NAME="E8:25_1"/>
<font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</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="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_1_1_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">c</font> = <font color="Maroon" title="c13">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:25_1_1_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c15">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E5:25_1_1_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="measure6.html#E24"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">d1</font> = <font color="Maroon" title="c15">d</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="E7:25_1_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E4:25_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E8:25_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c16">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<a NAME="E9:25_1_1_1"/>
<font color="Maroon" title="c16">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E4:25_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E10:25_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c16">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<a NAME="E11:25_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c16">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:25_1_1_1"/>
<font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</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="#E8:25_1_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E10:25_1_1_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c10">b</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> or <font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:25_1_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">c</font> = <font color="Maroon" title="c15">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">c1</font> = <font color="Maroon" title="c16">c</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="E6:25_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">y</font></span>)</span>
 
;<br/>
<a NAME="E7:25_1_1"/>
<font color="Maroon" title="c17">c1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c14">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E8:25_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<a NAME="E9:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c17">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T22" title="XXREAL_3:th.22">XXREAL_3:22</a></span>;<br/>
<a NAME="E10:25_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E11:25_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<a NAME="E12:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T22" title="XXREAL_3:th.22">XXREAL_3:22</a></span>;<br/>
<a NAME="E13:25_1_1"/>
<font color="Maroon" title="c17">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>;<br/>
<a NAME="E14:25_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E9:25_1_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:25_1_1"/>
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="measure6.html#E24"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:25_1"/>
<font color="Maroon" title="c4">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D3" title="MEASURE5:def.3">MEASURE5:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:25"/>
( <font color="Maroon" title="c2">x</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  implies <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c3">y</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:25_2"/><span class="lab"><font color="Green" title="E9">A12</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">x</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font> as   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a> ;<br/>
<a NAME="E4:25_2"/>
<font color="Maroon" title="c3">y</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T23" target="_self" title="MEASURE6:th.23">Th59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:25_2"/>
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:25_2"><span class="lab"><font color="Green" title="E9">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:25"/>
( <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  iff <font color="Maroon" title="c2">x</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
