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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:21_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E3:21_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 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#K1" title="RELSET_1:func.1">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>)
 <span class="kw">and </span><br/><a NAME="E4:21_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c4">Intf</font> is  <a href="limfunc1.html#V6" title="LIMFUNC1:attr.6">convergent_in-infty</a> 
 <span class="kw">and </span><br/><a NAME="E5:21_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="integr10.html#D8" target="_self" title="INTEGR10:def.8">Def8</a></span>;<br/>
<span class="kw">set </span><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="E6:21_1"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="limfunc1.html#T89" title="LIMFUNC1:th.89">LIMFUNC1:89</a></span>;<br/>
<a NAME="E7:21_1"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="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#K1" title="RELSET_1:func.1">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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">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/>

<span class="kw">assume </span><a NAME="E2:21_1_1"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></span>;<br/>
<a NAME="E4:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A12</font></span>: 
( <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> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="ref" href="integra5.html#D3" title="INTEGRA5:def.3">INTEGRA5:def 3</a></span>;<br/>
<a NAME="E5:21_1_1"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
( <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> &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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a>, <a class="ref" href="xxreal_1.html#T265" title="XXREAL_1:th.265">XXREAL_1:265</a></span>;<br/>
<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="integra6.html#T10" title="INTEGRA6:th.10">INTEGRA6:10</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:21_1"/>
 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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E1:21_1_2"/><span class="lab"><font color="Green" title="E21">A14</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T265" title="XXREAL_1:th.265">XXREAL_1:265</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:21_1_2"/><span class="lab"><font color="Green" title="E22">A15</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A16</font></span>: 
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a></span>;<br/>
<a NAME="E4:21_1_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="integra5.html#D3" title="INTEGRA5:def.3">INTEGRA5:def 3</a></span>;<br/>
<a NAME="E5:21_1_2"/><span class="kw">then </span>
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E1:21_1_2"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_1_2"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="integra6.html#T9" title="INTEGRA6:th.9">INTEGRA6:9</a>, <a class="ref" href="rfunct_1.html#T80" title="RFUNCT_1:th.80">RFUNCT_1:80</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:21_1"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="integr10.html#D6" target="_self" title="INTEGR10:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E10:21_1"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E5:21_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="limfunc1.html#T89" title="LIMFUNC1:th.89">LIMFUNC1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E9:21_1"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="integr10.html#D8" target="_self" title="INTEGR10:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:21"/>
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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