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<b>let </b><font color="Maroon">c<sub>1</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>2</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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E3:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="integra5.html#E3:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="integra5.html#E4:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1"/><b>then </b>
 <a href="integra5.html#K4" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="integra5.html#K3" target="_self">['</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="integra5.html#K3" target="_self">']</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="integra5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1"/>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E1:26_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="integra5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="integra5.html#K4" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="integra5.html#K3" target="_self">['</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="integra5.html#K3" target="_self">']</a></span>
 <b>by </b><i><a class="ref" href="integra5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E3:26_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E4:26_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green">E21</font></i></a>, <a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/><b>then </b><i><font color="Green">E26</font></i>:  <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra5.html#E1:26_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>
<br/>.= 
0

;<br/> <a href="integra5.html#K2" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
0
<b>by </b><i><a class="txt" href="integra5.html#E5:26_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="integra4.html#T6">INTEGRA4:6</a></i>
;<br/><b>hence </b><a NAME="E7:26_1_1_2"/>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra5.html#E1:26_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="integra5.html#E2:26_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="integra5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4" target="_self">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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