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<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</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> , <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>let </b><font color="Maroon">X</font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:27"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E7:27"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E8:27"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="fdiff_2.html#T20">FDIFF_2:20</a></i>;<br/>
<a NAME="E9:27"/>
( <font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">g</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<a NAME="E10:27"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E11:27"/><b>then </b>
( <font color="Maroon">f</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:27"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:27"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">X</font> <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="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>
<a NAME="E14:27"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">A</font> <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="E15:27"/><b>then </b>
( <font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">g</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">g</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E17:27"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:27"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E19:27"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E20:27"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E21:27"/><b>then </b><i><font color="Green">E82</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E22:27"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E23:27"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , , <a class="ref" href="integra4.html#T29">INTEGRA4:29</a></i>;<br/>
<a NAME="E24:27"/><b>then </b><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E25:27"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E26:27"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E27:27"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i/>;<br/>
<a NAME="E28:27"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E29:27"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="rfunct_1.html#T101">RFUNCT_1:101</a></i>;<br/>
<a NAME="E30:27"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E31:27"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E32:27"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E33:27"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i/>;<br/>
<a NAME="E34:27"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E35:27"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="rfunct_1.html#T101">RFUNCT_1:101</a></i>;<br/>
<a NAME="E36:27"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E37:27"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="fdiff_2.html#T20">FDIFF_2:20</a></i>;<br/>
<a NAME="E38:27"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , , <a class="ref" href="integra5.html#T1" target="_self">Th1</a>, <a class="ref" href="integra1.html#T59">INTEGRA1:59</a></i>;<br/>
<a NAME="E2:27_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:27"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="rfunct_1.html#T100">RFUNCT_1:100</a></i>;<br/>
<b>hence </b><a NAME="E2:27_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="fdiff_2.html#T20">FDIFF_2:20</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E95</font></i>:  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> = 
 <a href="integra5.html#K2" target="_self">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">A</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i>, , , , </i>
;<br/>
<a NAME="E41:27"/><i><font color="Green">E96</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:27_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:27_4"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E3:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E4:27_4"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:27_4"/><b>then </b><i><font color="Green">E98</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E6:27_4"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E7:27_4"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:27_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra5.html#E4:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:27"/>
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="integra5.html#T1" target="_self">Th1</a>, <a class="ref" href="integra1.html#T59">INTEGRA1:59</a></i>;<br/>
<a NAME="E43:27"/><b>then </b>
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" target="_self">integral</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" target="_self">integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E44:27"/>
 <a href="integra5.html#K2" target="_self">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" target="_self">integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="integra5.html#E1:27"><i><font color="Green">E47</font></i></a>, <a class="txt" href="integra5.html#E2:27"><i><font color="Green">E48</font></i></a></i>;<br/>


</div>
