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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">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">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">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="E1:26"/><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">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/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:26_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E3:26_1"/>
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra6.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:26_2"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:26_2"/>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>;<br/><a NAME="E4:26_2"/><b>then </b>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>;<br/><a NAME="E5:26_2"/><b>then </b>
 <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="hidden.html#R1">=</a>  <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">d</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E6:26_2"/>
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra6.html#T1" target="_self">Th1</a>, , <a class="txt" href="integra6.html#E1:26"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></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>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
