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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>









<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c5">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Maroon" title="c5">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> )
 ;<br/>

<div><i><font color="Green" title="E25">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font>
 ;<br/><a NAME="E2:26_1"/><b>then </b>
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E3:26_1"/>
( <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp;  <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp;  <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra6.html#E1:26"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="integra6.html#E1:26_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="integra6.html#E25"><i><font color="Green" title="E22">Lm5</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/><i><font color="Green" title="E26">A4</font></i>: 
not <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font>
 ;<br/><a NAME="E2:26_2"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:26_2"/>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c5">f</font>,<span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E1:26_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>;<br/><a NAME="E4:26_2"/><b>then </b>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E1:26_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>;<br/><a NAME="E5:26_2"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E6:26_2"/>
( <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp;  <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp;  <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra6.html#E1:26"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="integra6.html#E1:26_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="integra6.html#E2:26_2"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="integra6.html#E25"><i><font color="Green" title="E22">Lm5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
( <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp;  <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp;  <a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21" title="SEQ_1:func.21">abs</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra6.html#E2:26"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>


</div>
