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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:27"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:27"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E8:27"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra5.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E2:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fdiff_2.html#T20">FDIFF_2:20</a></i>;<br/>
<a NAME="E9:27"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E2:27"><i><font color="Green">E22</font></i></a>, <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">E29</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E3:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E11:27"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="integra5.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E12:27"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E4:27"><i><font color="Green">E24</font></i></a>, <a class="txt" href="integra5.html#E6:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="integra5.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E13:27"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E2:27"><i><font color="Green">E22</font></i></a>, <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">E31</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E3:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:27"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i><a class="ref" href="integra5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></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">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E10:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="integra5.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E18:27"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="integra5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E19:27"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra5.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E2:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="integra5.html#E3:27"><i><font color="Green">E23</font></i></a>, <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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i><a class="ref" href="integra5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E21:27"/><b>then </b><i><font color="Green">E36</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="integra5.html#E7:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="integra5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E23:27"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E12:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="integra5.html#E16:27"><i><font color="Green">E32</font></i></a>, <a class="txt" href="integra5.html#E18:27"><i><font color="Green">E34</font></i></a>, <a class="txt" href="integra5.html#E21:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra4.html#T29">INTEGRA4:29</a></i>;<br/>
<a NAME="E24:27"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="integra5.html#T4" target="_self">Th4</a></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra5.html#E14:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="integra5.html#E19:27"><i><font color="Green">E35</font></i></a>, <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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="integra5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E28:27"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></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">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra5.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="integra5.html#E17:27"><i><font color="Green">E33</font></i></a>, <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">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="integra5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E31:27"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></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">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra5.html#E14:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="integra5.html#E19:27"><i><font color="Green">E35</font></i></a>, <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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="integra5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E34:27"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></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">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra5.html#E7:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="integra5.html#E17:27"><i><font color="Green">E33</font></i></a>, <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">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="integra5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E37:27"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="integra5.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E2:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fdiff_2.html#T20">FDIFF_2:20</a></i>;<br/>
<a NAME="E38:27"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra5.html#E24:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra5.html#E28:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="integra5.html#E30:27"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra5.html#E34:27"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra5.html#E36:27"><i><font color="Green">E43</font></i></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="integra5.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="integra5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra5.html#E8:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="integra5.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:27"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra5.html#E29:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="integra5.html#E35:27"><i><font color="Green">E42</font></i></a>, <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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra5.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E2:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fdiff_2.html#T20">FDIFF_2:20</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E47</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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra5.html#E3:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="integra5.html#E37:27"><i><font color="Green">E44</font></i></a>, <a class="txt" href="integra5.html#E38:27"><i><font color="Green">E45</font></i></a>, <a class="txt" href="integra5.html#E39:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="integra5.html#T13" target="_self">Th13</a></i>
;<br/>
<a NAME="E41:27"/><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:27_4"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E2:27_4"><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">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></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">E50</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra5.html#E14:27"><i><font color="Green">E31</font></i></a>, <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">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra5.html#E5:27_4"><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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra5.html#E24:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra5.html#E28:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="integra5.html#E30:27"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra5.html#E34:27"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra5.html#E36:27"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#T59">INTEGRA1:59</a></i>;<br/>
<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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra5.html#T5" target="_self">Th5</a></i>
<br/>.= 
<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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>

;<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">c<sub>2</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>2</sub></font> <a href="seq_1.html#K8">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra5.html#E8:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="integra5.html#E40:27"><i><font color="Green">E47</font></i></a>, <a class="txt" href="integra5.html#E41:27"><i><font color="Green">E48</font></i></a></i>;<br/>


</div>
