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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>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="E1:33"/><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">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> &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><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:33_1"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <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> &amp;  <a href="integra5.html#K4">integral</a> <font color="Maroon">g</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">g</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> &amp;  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<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> &amp;  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>,<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="integra5.html#D5">INTEGRA5:def 5</a></i>;<br/><b>hence </b> <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="integra6.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span>

<br/>.= 
<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="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, , <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
<br/>.= 
<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="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, , <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
;<br/><br/><b>thus </b> <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, , <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">d</font>,<font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="integra6.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, , <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>

<br/>.= 
<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="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, , <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
(  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</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="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp;  <a href="integra5.html#K4">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</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="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra6.html#T3" target="_self">Th3</a></i>;<br/>


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