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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Olive" title="b2">r</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> holds <br/>( <font color="Olive" title="b1">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Olive" title="b1">f</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="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> holds <br/>( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</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="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> implies ( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</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="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> ) )</span><br/>





<b>assume </b><a NAME="E1:4"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</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="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> )</span><br/>

<a NAME="E2:4"/><b>then </b><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra4.html#T3" target="_self" title="INTEGRA4:th.3">Th3</a></i>;<br/>
<a NAME="E3:4"/><i><font color="Green" title="E6">A2</font></i>: 
(  <a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="integra4.html#T2" target="_self" title="INTEGRA4:th.2">Th2</a></i>;<br/>
<a NAME="E4:4"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131" title="FUNCT_2:th.131">FUNCT_2:131</a>, <a class="ref" href="rfunct_1.html#T78" title="RFUNCT_1:th.78">RFUNCT_1:78</a></i>;<br/>
<a NAME="E5:4"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default">1</span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T19" title="INTEGRA1:th.19">INTEGRA1:19</a></i>;<br/>
<a NAME="E6:4"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>
 
;<br/>
<a NAME="E7:4"/><b>then </b>
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> )
 
;<br/>
<a NAME="E8:4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#T14" title="INTEGRA1:th.14">INTEGRA1:14</a>, <a class="ref" href="integra1.html#T16" title="INTEGRA1:th.16">INTEGRA1:16</a></i>;<br/>
<a NAME="E9:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
;<br/>
<b>hence </b><a NAME="E10:4"/>
( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</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="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E3:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="#E4:4"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="integra2.html#T31" title="INTEGRA2:th.31">INTEGRA2:31</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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