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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 ;<br/>

<a NAME="E2:25"/>
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="integra5.html#D4">INTEGRA5:def 4</a></i>;<br/>
<a NAME="E3:25"/><b>then </b>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:25"/><b>then </b>
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:25"/>
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span> &amp;  <a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="integra5.html#K3">']</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i>, , , <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a>, <a class="ref" href="rfunct_1.html#T99">RFUNCT_1:99</a></i>;<br/>


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