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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> ;<br/><b>let </b><font color="Maroon">c</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">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">b</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>let </b><font color="Maroon">g</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="E1:22"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</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">g</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; <font color="Maroon">g</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; <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">g</font> )
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green">E34</font></i>: 
( <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> &amp; <font color="Maroon">g</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">g</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/>;<br/>
<a NAME="E3:22"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:22"/><b>then </b>
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:22"/><b>then </b><i><font color="Green">E35</font></i>: 
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</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; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</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">g</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:22"/>
<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> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:22"/><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; <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">g</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:22"/>
<font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</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>
 <b>by </b><i>, </i>;<br/>

<a NAME="E9:22"/>
<font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</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> <a href="subset_1.html#K9">/\</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>, <a class="ref" href="rfunct_1.html#T100">RFUNCT_1:100</a></i>;<br/>
<b>hence </b><a NAME="E10:22"/>
<font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</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>
 ;<br/>


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