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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Olive" title="b1">b</font> holds <br/>for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Olive" title="b1">b</font> &amp;  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">b</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">b</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font> implies for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span> )</span><br/>

<a NAME="E2:21"/>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">Intf</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:21_1"/><i><font color="Green" title="E14">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">Intf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">Intf</font> holds <br/><font color="Maroon" title="c4">Intf</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">b</font> ) &amp; <font color="Maroon" title="c4">Intf</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> &amp;  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <font color="Maroon" title="c4">Intf</font> )
 <b>by </b><i><a class="ref" href="integr10.html#D8" target="_self" title="INTEGR10:def.8">Def8</a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">Intfg</font> = <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font>;<br/>
<a NAME="E3:21_1"/><i><font color="Green" title="E15">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> holds <br/>( <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> implies ( <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) )</span><br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )</span><br/>

<a NAME="E2:21_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <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="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra5.html#D4" title="INTEGRA5:def.4">INTEGRA5:def 4</a>, <a class="txt" href="#E1:21_1_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E3:21_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T265" title="XXREAL_1:th.265">XXREAL_1:265</a></i>;<br/>
<a NAME="E4:21_1_1"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="#E2:21_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:21_1_1"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 
<b>by </b><i><a class="txt" href="#E1:21_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:21_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 <b>by </b><i><a class="txt" href="#E5:21_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="rfunct_1.html#T97" title="RFUNCT_1:th.97">RFUNCT_1:97</a>, <a class="txt" href="#E4:21_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="integra6.html#T9" title="INTEGRA6:th.9">INTEGRA6:9</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>
<a NAME="E4:21_1"/><i><font color="Green" title="E16">A8</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:21_1_2"/><i><font color="Green" title="E17">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">b</font></span><br/>

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font> )</span><br/>

<b>assume </b><a NAME="E2:21_1_2"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E3:21_1_2"/><b>then </b><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="#E1:21_1_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></i>;<br/>
<a NAME="E4:21_1_2"/><i><font color="Green" title="E20">A12</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <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="c6">x</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra5.html#D4" title="INTEGRA5:def.4">INTEGRA5:def 4</a>, <a class="txt" href="#E3:21_1_2"><i><font color="Green" title="E19">A11</font></i></a></i>;<br/>
<a NAME="E5:21_1_2"/><i><font color="Green" title="E21">A13</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T265" title="XXREAL_1:th.265">XXREAL_1:265</a></i>;<br/>
<a NAME="E6:21_1_2"/><i><font color="Green" title="E22">A14</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 
<b>by </b><i><a class="txt" href="#E3:21_1_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Intf</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a>, <a class="txt" href="#E2:21_1_2"><i><font color="Green" title="E18">A10</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E1:21_1_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="#E2:21_1_2"><i><font color="Green" title="E18">A10</font></i></a></i>
<br/>.= 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="txt" href="#E5:21_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E4:21_1_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="txt" href="#E6:21_1_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="integra6.html#T10" title="INTEGRA6:th.10">INTEGRA6:10</a>, <a class="txt" href="#E3:21_1_2"><i><font color="Green" title="E19">A11</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>
<a NAME="E5:21_1"/><i><font color="Green" title="E17">A15</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>
 
<b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="limfunc1.html#T124" title="LIMFUNC1:th.124">LIMFUNC1:124</a></i>;<br/>
<a NAME="E6:21_1"/><i><font color="Green" title="E18">A16</font></i>: 
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T124" title="LIMFUNC1:th.124">LIMFUNC1:124</a>, <a class="txt" href="#E2:21_1"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:21_1"/>
<font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="#E3:21_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E4:21_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E5:21_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span></span><br/>

<b>hence </b><a NAME="E8:21_1"/>
 <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:21_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="#E5:21_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="txt" href="#E6:21_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="integr10.html#D8" target="_self" title="INTEGR10:def.8">Def8</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>
<b>hence </b><a NAME="E3:21"/>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">b</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
