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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</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>   st <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">f</font> <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>  holds <br/>(  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <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>  implies (  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">A</font> ) )</span><br/>



<a NAME="E1:8"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T62" title="RFUNCT_1:th.62">RFUNCT_1:62</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8"/>
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> or  not <font color="Maroon" title="c2">f</font> <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>  or (  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">A</font> ) )</span><br/>

<a NAME="E3:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integra6.html#E7"><span class="lab"><font color="Green" title="E7">Lm1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:8"/>
( <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">A</font> )</span><br/>

<a NAME="E5:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T9" title="INTEGRA5:th.9">INTEGRA5:9</a>, <a class="ref" href="integra5.html#D2" title="INTEGRA5:def.2">INTEGRA5:def 2</a></span>;<br/>
<a NAME="E6:8"/><span class="kw">then </span>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="integra4.html#T23" title="INTEGRA4:th.23">INTEGRA4:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:8"/>
(  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="integra4.html#T23" title="INTEGRA4:th.23">INTEGRA4:23</a>, <a class="ref" href="integra5.html#D2" title="INTEGRA5:def.2">INTEGRA5:def 2</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/>


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