<?xml version="1.0"?>
<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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/>( ( for <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies ( ( for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">Intf</font> =  <a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E16">A3</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="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr10.html#D12" target="_self" title="INTEGR10:def.12">Def12</a></span>
;<br/>
<a NAME="E4:27"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">s</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> ) )</span><br/>

<a NAME="E1:27_2"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T24" title="RFUNCT_1:th.24">RFUNCT_1:24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:27_2"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )</span><br/>

<a NAME="E3:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:27_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E1:27_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="integr10.html#T9" target="_self" title="INTEGR10:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:27"/>
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</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="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">s</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s</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="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_3"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c5">s</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="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:27_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c5">s</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"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="integr10.html#D12" target="_self" title="INTEGR10:def.12">Def12</a></span>;<br/>
<span class="kw">thus </span> <a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integr10.html#K3" title="INTEGR10:func.3">infty_ext_right_integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E1:27_3"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_3"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="integr10.html#D12" target="_self" title="INTEGR10:def.12">Def12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:27"/>
( ( for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K6" title="INTEGR10:func.6">exp*-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="integr10.html#R3" title="INTEGR10:pred.3">is_+infty_ext_Riemann_integrable_on</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="integr10.html#K7" title="INTEGR10:func.7">One-sided_Laplace_transform</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="integr10.html#D12" target="_self" title="INTEGR10:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
