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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">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>set </b><font color="Maroon">F</font> = <font color="Maroon">f</font> <a href="integra5.html#K1">||</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/>
<b>set </b><font color="Maroon">G</font> = <font color="Maroon">g</font> <a href="integra5.html#K1">||</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/>
<b>assume </b><a NAME="E1:13"/><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; <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">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> )
 ;<br/>

<a NAME="E2:13"/><b>then </b>
(  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<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;  <a href="integra5.html#K4">integral</a> <font color="Maroon">g</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2">integral</a> <font color="Maroon">g</font>,<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;  <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">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2">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>,<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;  <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">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2">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>,<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><a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:13"/>
(  <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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


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