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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</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;  <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="c2">g</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> &amp; <font color="Maroon" title="c2">g</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/>( <font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Olive" title="b1">b</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">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="c3">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;  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c3">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="c2">g</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="c3">b</font> &amp; <font color="Maroon" title="c2">g</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c3">b</font> implies ( <font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Maroon" title="c3">b</font></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:20"/><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="c3">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;  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c3">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="c2">g</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="c3">b</font> &amp; <font color="Maroon" title="c2">g</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Maroon" title="c3">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="E3:20"/><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="c3">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="c3">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="c3">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="#E1:20"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">Intg</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="E5:20"/><i><font color="Green" title="E15">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">Intg</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="c3">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="c5">Intg</font> holds <br/><font color="Maroon" title="c5">Intg</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="c2">g</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font> ) &amp; <font color="Maroon" title="c5">Intg</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="c2">g</font>,<font color="Maroon" title="c3">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="c5">Intg</font> )
 <b>by </b><i><a class="ref" href="integr10.html#D8" target="_self" title="INTEGR10:def.8">Def8</a>, <a class="txt" href="#E1:20"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Intfg</font> = <font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font>;<br/>
<a NAME="E6:20"/><i><font color="Green" title="E16">A4</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="c3">b</font> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c3">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="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> implies ( <font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">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="E1:20_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">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="E2:20_1"/><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="c7">a</font>,<font color="Maroon" title="c3">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="c7">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra5.html#D4" title="INTEGRA5:def.4">INTEGRA5:def 4</a>, <a class="txt" href="#E1:20_1"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<a NAME="E3:20_1"/><i><font color="Green" title="E19">A7</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c3">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="c3">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="E4:20_1"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c3">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="#E2:20_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E1:20"><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="E5:20_1"/><i><font color="Green" title="E21">A9</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c3">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="c2">g</font>
 
<b>by </b><i><a class="txt" href="#E2:20_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E3:20_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="#E1:20"><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="E6:20_1"/><i><font color="Green" title="E22">A10</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="c7">a</font>,<font color="Maroon" title="c3">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="c7">a</font>,<font color="Maroon" title="c3">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="#E1:20_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="txt" href="#E1:20"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<a NAME="E7:20_1"/><i><font color="Green" title="E23">A11</font></i>: 
( <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">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="#E1:20_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="txt" href="#E1:20"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<a NAME="E8:20_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="#E6:20_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="#E7:20_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="rfunct_1.html#T100" title="RFUNCT_1:th.100">RFUNCT_1:100</a></i>;<br/>
<b>hence </b><a NAME="E9:20_1"/>
( <font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</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="c7">a</font>,<font color="Maroon" title="c3">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="#E6:20_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="#E7:20_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="#E4:20_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="#E5:20_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="integra6.html#T11" title="INTEGRA6:th.11">INTEGRA6:11</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="E7:20"/><i><font color="Green" title="E17">A12</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</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="c3">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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>thus </b><i><font color="Green" title="E18">A13</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">Intf</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">Intg</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
 <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="#E3:20"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E5:20"><i><font color="Green" title="E15">A3</font></i></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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font></span><br/>

<b>let </b><font color="Maroon" title="c7">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="c7">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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">b</font> )</span><br/>

<b>assume </b><a NAME="E2:20_2"/><i><font color="Green" title="E19">A14</font></i>: 
<font color="Maroon" title="c7">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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E3:20_2"/><b>then </b><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E1:20_2"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></i>;<br/>
<a NAME="E4:20_2"/><i><font color="Green" title="E21">A16</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">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="c7">x</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra5.html#D4" title="INTEGRA5:def.4">INTEGRA5:def 4</a>, <a class="txt" href="#E3:20_2"><i><font color="Green" title="E20">A15</font></i></a></i>;<br/>
<a NAME="E5:20_2"/><i><font color="Green" title="E22">A17</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">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="c3">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="E6:20_2"/><b>then </b><i><font color="Green" title="E23">A18</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">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="#E4:20_2"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="#E1:20"><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="E7:20_2"/><i><font color="Green" title="E24">A19</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">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="c2">g</font>
 
<b>by </b><i><a class="txt" href="#E4:20_2"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="#E5:20_2"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="#E1:20"><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="E8:20_2"/><i><font color="Green" title="E25">A20</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="c7">x</font>,<font color="Maroon" title="c3">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="c7">x</font>,<font color="Maroon" title="c3">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="#E3:20_2"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="txt" href="#E1:20"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<a NAME="E9:20_2"/><i><font color="Green" title="E26">A21</font></i>: 
( <font color="Maroon" title="c2">g</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="c7">x</font>,<font color="Maroon" title="c3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Maroon" title="c2">g</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="c7">x</font>,<font color="Maroon" title="c3">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="#E3:20_2"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="txt" href="#E1:20"><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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<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="c7">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Intg</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a>, <a class="txt" href="#E2:20_2"><i><font color="Green" title="E19">A14</font></i></a></i>
<br/>.= 
<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="c7">x</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Intg</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:20"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E1:20_2"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="#E2:20_2"><i><font color="Green" title="E19">A14</font></i></a></i>
<br/>.= 
<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="c7">x</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:20"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E1:20_2"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="#E2:20_2"><i><font color="Green" title="E19">A14</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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="#E6:20_2"><i><font color="Green" title="E23">A18</font></i></a>, <a class="txt" href="#E7:20_2"><i><font color="Green" title="E24">A19</font></i></a>, <a class="txt" href="#E8:20_2"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="#E9:20_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="integra6.html#T12" title="INTEGRA6:th.12">INTEGRA6:12</a>, <a class="txt" href="#E3:20_2"><i><font color="Green" title="E20">A15</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="E8:20"/><i><font color="Green" title="E18">A22</font></i>: 
( <font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> &amp;  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Maroon" title="c3">b</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<a NAME="E1:20_3"/><i><font color="Green" title="E19">A23</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   ex <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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">  ex <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Olive" title="b1">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_3_1_1"/>
( <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> or  not <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_3_1_1_1"><b>suppose </b></a><a NAME="E1:20_3_1_1_1"/><i><font color="Green" title="E20">A24</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Olive" title="b1">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c3">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c8">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )</span><br/><b>thus </b><a NAME="E3:20_3_1_1_1"/>
( <font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c8">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:20_3_1_1_1"><i><font color="Green" title="E20">A24</font></i></a>, <a class="txt" href="#E7:20"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_3_1_1_2"><b>suppose </b></a><a NAME="E1:20_3_1_1_2"/><i><font color="Green" title="E20">A25</font></i>: 
not <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Olive" title="b1">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><b>take </b>
<font color="Maroon" title="c8">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c8">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )</span><br/><a NAME="E3:20_3_1_1_2"/><i><font color="Green" title="E21">A26</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E4:20_3_1_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="#E1:20_3_1_1_2"><i><font color="Green" title="E20">A25</font></i></a>, <a class="txt" href="#E3:20_3_1_1_2"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:20_3_1_1_2"/>
( <font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c8">g</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="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:20_3_1_1_2"><i><font color="Green" title="E21">A26</font></i></a>, <a class="txt" href="#E7:20"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E2:20_3"/>
<font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font> is <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T126" title="LIMFUNC1:th.126">LIMFUNC1:126</a>, <a class="txt" href="#E3:20"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E5:20"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E1:20_3"><i><font color="Green" title="E19">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Maroon" title="c3">b</font></span>)</span></span><br/>

<b>thus </b><a NAME="E3:20_3"/>
 <a href="limfunc1.html#K6" title="LIMFUNC1:func.6">lim_in-infty</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Intf</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">Intg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="limfunc1.html#T126" title="LIMFUNC1:th.126">LIMFUNC1:126</a>, <a class="txt" href="#E3:20"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E5:20"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E1:20_3"><i><font color="Green" title="E19">A23</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>
<b>thus </b><a NAME="E9:20"/>
<font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font> <a href="integr10.html#R4" title="INTEGR10:pred.4">is_-infty_ext_Riemann_integrable_on</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="#E6:20"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="#E7:20"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="#E8:20"><i><font color="Green" title="E18">A22</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="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Maroon" title="c3">b</font></span>)</span></span><br/>

<b>hence </b><a NAME="E10:20"/>
 <a href="integr10.html#K4" title="INTEGR10:func.4">infty_ext_left_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">g</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:20"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="#E8:20"><i><font color="Green" title="E18">A22</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/>


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