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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">b</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <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>   st <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">c</font> &amp; <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b1">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="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Olive" title="b1">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="c4">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="c4">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b1">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="c2">c</font>,<font color="Maroon" title="c3">d</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Olive" title="b1">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="c2">c</font>,<font color="Maroon" title="c3">d</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="c2">c</font>,<font color="Maroon" title="c3">d</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> )</span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">c</font> &amp; <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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="c4">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="c4">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="c4">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> implies ( <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="c2">c</font>,<font color="Maroon" title="c3">d</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="c2">c</font>,<font color="Maroon" title="c3">d</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="c2">c</font>,<font color="Maroon" title="c3">d</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E20">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">c</font> &amp; <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> )
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E21">A2</font></i>: 
( <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="c4">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="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> )
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green" title="E22">A3</font></i>: 
<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="c4">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c2">c</font>,<font color="Maroon" title="c3">d</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="c2">c</font>,<font color="Maroon" title="c3">d</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="c2">c</font>,<font color="Maroon" title="c3">d</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<a NAME="E4:21"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:21"/><b>then </b><i><font color="Green" title="E23">A4</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="c4">b</font> &amp; <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="integra6.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:21"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c2">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="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Maroon" title="c3">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="c2">c</font>,<font color="Maroon" title="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> )
 
<b>by </b><i><a class="txt" href="integra6.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="integra6.html#E20"><i><font color="Green" title="E18">Lm3</font></i></a></i>;<br/>
<a NAME="E7:21"/><i><font color="Green" title="E25">A6</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c4">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> <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="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 
<b>by </b><i><a class="txt" href="integra6.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="integra6.html#E5:21"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="integra6.html#E20"><i><font color="Green" title="E18">Lm3</font></i></a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c4">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:21"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:21"/>
( <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="c2">c</font>,<font color="Maroon" title="c4">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="c2">c</font>,<font color="Maroon" title="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> )
 
<b>by </b><i><a class="txt" href="integra6.html#E2:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="integra6.html#E3:21"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="integra6.html#E5:21"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="integra6.html#E6:21"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="integra6.html#E7:21"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="integra6.html#T17" target="_self" title="INTEGRA6:th.17">Th17</a>, <a class="ref" href="rfunct_1.html#T91" title="RFUNCT_1:th.91">RFUNCT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E10:21"/>
<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="c2">c</font>,<font color="Maroon" title="c3">d</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 <b>by </b><i><a class="txt" href="integra6.html#E5:21"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="integra6.html#E6:21"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="integra6.html#E8:21"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="integra6.html#T17" target="_self" title="INTEGRA6:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c2">c</font>,<font color="Maroon" title="c3">d</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="c2">c</font>,<font color="Maroon" title="c3">d</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<a NAME="E11:21"/>
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 
<b>by </b><i><a class="txt" href="integra6.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="integra6.html#E20"><i><font color="Green" title="E18">Lm3</font></i></a></i>;<br/>
<a NAME="E12:21"/><b>then </b><i><font color="Green" title="E27">A8</font></i>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 
<b>by </b><i><a class="txt" href="integra6.html#E7:21"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:21"/>
<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="c2">c</font>,<font color="Maroon" title="c3">d</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 <b>by </b><i><a class="txt" href="integra6.html#E2:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="rfunct_1.html#T91" title="RFUNCT_1:th.91">RFUNCT_1:91</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span><br/>

<b>thus </b><a NAME="E14:21"/>
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="integra6.html#E3:21"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="integra6.html#E12:21"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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